Cremona transformations and rational parametrizations inspired by Hodge theory by Kuan-Wen Lai B. Sc., National Taiwan University; Taipei, Taiwan, 2009 M. Sc., National Taiwan University; Taipei, Taiwan, 2011 A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Mathematics at Brown University Providence, Rhode Island May 2018 c Copyright 2018 by Kuan-Wen Lai This dissertation by Kuan-Wen Lai is accepted in its present form by the Department of Mathematics as satisfying the dissertation requirement for the degree of Doctor of Philosophy. Date Brendan Hassett, Ph. D., Advisor Recommended to the Graduate Council Date Dan Abramovich, Ph. D., Reader Date Joseph H. Silverman, Ph. D., Reader Approved by the Graduate Council Date Andrew G. Campbell Dean of the Graduate School iii Curriculum Vitae Kuan-Wen Lai grew up in Taichung, Taiwan. He earned a Bachelor of Science degree in physics from National Taiwan University in 2005 and a Master of Science degree in mathematics from the same university in 2009. Then he served in the military for eleven months, where he spent the last six month in Kinmen as a second lieutenant. After being a research assistant in Taida Institute for Mathematical Sciences for one year, he came to the U.S. in 2013 and started his PhD program at Rice University. He transferred to Brown University in 2015 and completed his PhD in algebraic geometry in May 2018 under the supervision of Brendan Hassett. Publication list (1) Kuan-Wen Lai, New cubic fourfolds with odd degree unirational parametrizations. arXiv:1606.03853. Algebra and Number Theory, 11 (2017), 1597-1626. (2) Brendan Hassett and Kuan-Wen Lai, Cremona transformations and derived equiva- lences of K3 surfaces. arXiv:1612.07751. Compositio Mathematica, to appear. iv Preface The geometry of K3 surfaces and cubic fourfolds is deeply connected with their Hodge structures. Can we realize this connection via birational geometry? Two topics concern us here: derived equivalences of K3 surfaces, and the rationality problem of cubic fourfolds. How does the derived category of coherent sheaves on an algebraic variety determine the underlying geometry? For K3 surfaces, derived equivalences can be interpreted via iso- morphisms of Hodge structures. Classically, K3 surfaces with the same derived category can be realized through various geometric constructions. We introduce a new construction via Cremona transformations in Chapter 1. When is a cubic fourfold birational to P4 ? In some special cases, the Hodge struc- ture of a cubic fourfold may contain a substructure from a K3 surface. In Chapter 3, we use this relation to produce rational scrolls inside the cubics for two different examples. One of them can be proved to be rational, while the other is proved to admit unirational parametrizations of odd degree. The preliminary results on rational scrolls are organized in Chapter 2. I would like to thank my advisor, Brendan Hassett, for his guidance throughout my graduate life in Rice University and Brown University. I am also grateful to Kenneth Ascher and Dori Bejleri for their help in preparing for this thesis. My special thanks go to Wen-Yao Lee — thank you for being with me all the time. v Contents Introduction 1 1 Cremona transformations and K3 surfaces 4 1.1 Cremona transformations with nodal base loci . . . . . . . . . . . . . . . 7 1.1.1 Resolving the rational map . . . . . . . . . . . . . . . . . . . . . 8 1.1.2 Computing the intersection numbers . . . . . . . . . . . . . . . . 10 1.2 K3 surfaces of degree 12 and the construction . . . . . . . . . . . . . . . 13 1.2.1 Derived equivalences of K3 surfaces . . . . . . . . . . . . . . . . 13 1.2.2 K3 surfaces of degree 12 . . . . . . . . . . . . . . . . . . . . . . 15 1.2.3 Linear sections on the spinor tenfold . . . . . . . . . . . . . . . . 16 1.2.4 The data for our example . . . . . . . . . . . . . . . . . . . . . . 18 1.2.5 Proof of Theorem 1.6 . . . . . . . . . . . . . . . . . . . . . . . . 19 1.3 Derived equivalence via Cremona transformation . . . . . . . . . . . . . 21 1.3.1 The middle cohomology of X . . . . . . . . . . . . . . . . . . . 22 1.3.2 The discriminant groups . . . . . . . . . . . . . . . . . . . . . . 25 1.3.3 Proofs of Theorem 1.10 and its Corollary . . . . . . . . . . . . . 31 1.3.4 Locus contructed by the map . . . . . . . . . . . . . . . . . . . . 33 1.3.5 Zero divisors in the Grothendieck ring . . . . . . . . . . . . . . . 35 vi 1.4 Exclusion of alternative constructions . . . . . . . . . . . . . . . . . . . 38 1.4.1 Extracting Diophantine equations . . . . . . . . . . . . . . . . . 38 1.4.2 Enumeration of combinatorial cases . . . . . . . . . . . . . . . . 39 1.4.3 Exclusion of cases . . . . . . . . . . . . . . . . . . . . . . . . . 40 1.4.4 Geometric analysis of the remaining case . . . . . . . . . . . . . 44 2 Nodal rational scrolls and the Hilbert schemes 51 2.1 Rational scrolls and the singularities . . . . . . . . . . . . . . . . . . . . 52 2.1.1 Hirzebruch surfaces . . . . . . . . . . . . . . . . . . . . . . . . 52 2.1.2 Rational normal scrolls . . . . . . . . . . . . . . . . . . . . . . . 54 2.1.3 Rational scrolls . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 2.1.4 Singularities on rational scrolls . . . . . . . . . . . . . . . . . . . 59 2.2 Constructing nodal scrolls in P5 . . . . . . . . . . . . . . . . . . . . . . 63 2.2.1 Plane k-chains . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 2.2.2 Control the number of nodes . . . . . . . . . . . . . . . . . . . . 65 2.2.3 Proof of Proposition 2.12 . . . . . . . . . . . . . . . . . . . . . . 70 2.3 Hilbert schemes of rational scrolls . . . . . . . . . . . . . . . . . . . . . 73 2.3.1 Parametrizing the Hilbert schemes . . . . . . . . . . . . . . . . . 74 2.3.2 Projections that produce one singularity . . . . . . . . . . . . . . 76 2.3.3 The geometry of the variety σ(l, l0 ) . . . . . . . . . . . . . . . . 78 2.3.4 Families of the projections . . . . . . . . . . . . . . . . . . . . . 82 2.3.5 Proof of Theorem 2.17 . . . . . . . . . . . . . . . . . . . . . . . 84 3 Rationality and unirationality of cubic fourfolds 89 3.1 Cubic fourfolds with associated K3 surfaces . . . . . . . . . . . . . . . . 90 3.1.1 Pfaffian cubic fourfolds . . . . . . . . . . . . . . . . . . . . . . . 90 vii 3.1.2 Special cubic fourfolds . . . . . . . . . . . . . . . . . . . . . . . 93 3.1.3 Associated K3 surfaces . . . . . . . . . . . . . . . . . . . . . . . 94 3.1.4 Cycles on the Hilbert square . . . . . . . . . . . . . . . . . . . . 95 3.1.5 Scrolls from the Hilbert square . . . . . . . . . . . . . . . . . . . 97 3.2 Rationality of cubic fourfolds of discriminant 26 . . . . . . . . . . . . . . 99 3.2.1 Septic scrolls with three nodes . . . . . . . . . . . . . . . . . . . 99 3.2.2 The birational transformation . . . . . . . . . . . . . . . . . . . 102 3.2.3 The rational parametrization . . . . . . . . . . . . . . . . . . . . 106 3.3 Unirational parametrizations of odd degree . . . . . . . . . . . . . . . . 108 3.3.1 Degree-9 scrolls with eight nodes . . . . . . . . . . . . . . . . . 110 3.3.2 Unirational parametrizations of degree 13 . . . . . . . . . . . . . 111 Bibliography 114 viii Introduction Cubic fourfolds are complex smooth cubic hypersurfaces in P5 . Given a cubic fourfold ∼ X, can we find a birational map P4 d X, i.e. is X rational? It is known that such a parametrization is not allowed for a cubic threefold Y ⊂ P4 . Indeed, resolving a birational ∼ map P3 d Y by blowing up induces a Hodge isomorphism H 3 (Y, Z)  H 1 (C, Z) for some curve C ⊂ P3 . However, this is impossible since H 3 (Y, Z) contains a component not from a curve according to Clemens and Griffiths [CG72]. By virtue of this result— although no example is found so far—it is expected that a general cubic fourfold is irra- tional. In contrast, some special cubic fourfolds are shown to be rational. Example 0.1. Here is a rational example. Let (u, v, w, x, y, z) be homogeneous coordinates on P5 . Consider the two disjoint planes P1 := {x = y = z = 0}, P2 := {u = v = w = 0}. Then X := {ux2 + vy2 + wz2 = u2 x + v2 y + w2 z} is a cubic fourfold containing P1 and P2 . Given p1 ∈ P1 and p2 ∈ P2 in general positions, 1 2 the line spanned by them intersects X in exactly one other point ρ(p1 , p2 ). This defines a birational map (0.1) ρ : P1 × P2 ∼ / X, so X is birational to P4 . Furthermore, the base locus S ⊂ P1 × P2 of ρ consists of the pairs (p1 , p2 ) which spans a line contained in X. Consider F1,2 := ux2 + vy2 + wz2 , F2,1 := u2 x + v2 y + w2 z as hypersurfaces of bidegrees (1, 2) and (2, 1) in P1 × P2 , respectively. Then the above condition is equivalent to F1,2 (p1 , p2 ) = F2,1 (p1 , p2 ) = 0. Therefore, S is a complete intersection of F1,2 and F2,1 and thus a K3 surface. We say a cubic fourfold X has an associated K3 surface S if there is an embedding  T (S )  / H 4 (X, Z) respecting the Hodge structures, where T (S ) is the transcendental lattice of S , defined as the sublattice of H 2 (S , Z) orthogonal to the algebraic classes. One can obtain this relation in the above example by blowing up the base locus of the rational map (0.1). There are examples of cubic fourfolds admitting multiple associated K3 surfaces. If they are rational, then we expect that there exists rational fourfolds P and P0 and birational maps ρ ρ0 P / Xo P0 ∼ ∼ such that the base loci of ρ and ρ0 are K3 surfaces. Note that the composition (ρ0 )−1 ◦ ρ induces a birational automorphism of P4 . This motivates the following question: 3 Question 0.2. Does there exist a birational automorphism of P4 which can be resolved by blowing up a K3 surface? Unfortunately, there is no such example due to the classification of Crauder and Katz [CK89], where they prove that there are only two classes of birational automorphisms of P4 undefined along smooth irreducible surfaces and both are non-K3. However, this situation can be remedied by allowing singularities on the base locus. In Chapter 1, we exhibit such an example as well as several interesting conclusions derived from its construction. Kuznetsov [Kuz10, AT14] has conjectured that we can predict the rationality of cubic fourfolds from the existence of associated K3 surfaces. The first such example was estab- lished by Morin and Fano [Mor40, Fan43] in the 1940s. Two new cases are proved decades after by Russo and Staglian`o [RS17]. On the other hand, it is well-known that every com- plex smooth cubic hypersurface X ⊂ Pn+1 admits a degree-two unirational parametriza- tion, i.e. a rational double cover Pn d X. Hence it is reasonable to ask Question 0.3. Given a cubic fourfold X, does it admit a unirational parametrization of odd degree? The rationality problem will be discussed in Chapter 3, where we go through one of Russo and Staglian`o’s examples in detail, as well as give a new example admitting unirational parametrizations of odd degree. In each case, the parametrization is built upon the existence of certain rational scroll R ⊂ X which arise from the isomorphism F1 (X)  S [2] between the variety of lines on X and the Hilbert scheme of two points on the associated K3 surface S . Chapter 1 Cremona transformations and K3 surfaces Let X be a smooth complex projective variety of dimension n. Assume that it is rational, that is, admits a birational map ρ : Pn ∼ / X. When n = 1, ρ extends to an isomorphism; if n = 2, ρ is resolved by blowing up points in P2 . In general dimensions, resolving ρ requires blowing up various subvarieties of Pn — to what extent are these determined by X? The case of threefolds was analyzed by Clemens and Griffiths [CG72]. We may as- sume that ρ (resp. ρ−1 ) is resolved by blowing up a finite number of points and nonsingular irreducible curves; let C1 , . . . , Ck (resp. D1 , . . . , Dl ) denote those of positive genus. Com- paring the Hodge structures on middle cohomology groups using the blow-up formula, we obtain an isomorphism of principally polarized abelian varieties: J(C1 ) × · · · × J(Ck ) ' J(X) × J(D1 ) × · · · × J(Dl ). 4 5 The factors are Jacobians of curves and the intermediate Jacobian of X. Principally polar- ized abelian varieties admit unique decompositions into irreducible factors and the Jaco- bian of a curve is irreducible with respect to the natural polarization. It follows that J(X) can be expressed as a product of Jacobians of curves Ci1 , . . . , Cit , {i1 , . . . , it } ⊂ {1, . . . , k}, and these curves are determined up to isomorphism by the Torelli Theorem. Therefore, we focus on fourfolds and their middle cohomologies. Suppose that a smooth projective surface Σ is contained in the base locus of ρ. The blow-up formula gives a homomorphism of Hodge structures β : H 2 (Σ, Z)(−1) → H 4 (X, Z); can we recover Σ from the image of β? Keeping track of divisor classes of Σ is complicated, as they might disappear under subsequent blow-downs. Thus all we can expect to recover is the transcendental cohomology T (Σ) ⊂ H 2 (Σ, Z). Mukai and Orlov [Orl97] have shown that K3 surfaces offer many examples of non- birational surfaces R and Rˆ with T (R) ' T (R) ˆ as integral Hodge structures. These are explained through the notion of derived equivalence. This raises a question: Question 1.1. Let R and Rˆ be derived equivalent K3 surfaces. Do there exist smooth ˆ and birational maps projective fourfolds X, P, and P, ρ ρˆ P / Xo ˆ P, such that R and Rˆ are birational to components of the base loci of % and %ˆ respectively, and the induced homomorphisms β βˆ H 2 (R, Z)(−1) / H 4 (X, Z) o ˆ Z)(−1), H 2 (R, 6 ˆ induce an isomorphism T (R) ' T (R)? In other words, are derived equivalences of K3 surfaces induced by birational maps? It makes sense to start with the case where P ' Pˆ ' P4 . Are derived equivalences of K3 sur- faces induced by Cremona transformations, that is, birational automorphisms of projective spaces? The following theorem gives a positive answer for degree-12 K3 surfaces. Theorem 1.2. For a general K3 surface R of degree 12, there exists a Cremona transfor- mation f : P4 ∼ / P4 such that 1. the base locus of f is birational to R; ˆ 2. the base locus of the inverse f −1 is birational to a degree-12 K3 surface R; 3. R and Rˆ are derived equivalent but not isomorphic to each other. Moreover, the construction provides (i) a birational map R[3] ∼ / Rˆ [3] between the Hilbert schemes of three points; (ii) a relation in the Grothendieck ring K0 (Var/C) of varieties: ˆ ([R] − [R])L = 0, where L = [A1 ] is the class of the affine line. This relation is nontrivial in the sense ˆ that [R] , [R]. However, the base loci of Cremona transformations are highly constrained. According to Crauder and Katz [CK89], the Cremona transformation of P4 which can be resolved by blowing up along a smooth and irreducible surface S ⊂ P4 occurs as one of the following two cases: 7 1. S is a quintic elliptic scroll S = PC (E), where C is an elliptic curve and E is a rank two vector bundle with e = − deg( 2 E) = −1. V 2. S is a degree 10 determinantal surface given by the vanishing of the 4 × 4 minors of a 4 × 5 matrix of linear forms. The base locus of our rational map is singular along three nodes, i.e. points where the surface has two smooth branches meeting transversally. In addition to the two smooth cases, this is the only example when S is allowed to be nodal, which forbids us to realize derived equivalences of higher degree K3 surfaces in a similar way. Section 1.1 presents preliminary results on Cremona transformations with nodal base loci. We review the basic facts on K3 surfaces in Section 1.2, where we also give the construction of our rational map. We verify the nontrivial derived equivalence and apply the result to study the Grothendieck ring in Section 1.3. In Section 1.4, we show that these constructions do not admit obvious extensions through a generalization of the classifica- tion of Crauder and Katz; the underlying computations are also used to analyze the maps defined in Section 1.2. 1.1 Cremona transformations with nodal base loci A Cremona transformation is a birational automorphism of Pn which is not a linear trans- formation. Its base locus is the subscheme where the map is undefined. We consider the Cremona transformation f : P4 ∼ / P4 with base locus resolved by blowing up an irreducible surface S ⊂ P4 , with singular locus consisting of nodes, which means a point where the surface has two smooth branches 8 meeting transversally. This section describes a resolution of f . The terminology and notation developed during the process will be used throughout this chapter. 1.1.1 Resolving the rational map Suppose S has δ nodes which form a subset ∆ ⊂ P4 . The blowup of P4 along S can be factored as follows: 1. Blow up P4 along ∆, introducing δ exceptional divisors E1 , ..., Eδ isomorphic to P3 . Let P denote the resulting fourfold and S 0 the proper transform of S , which is now smooth. 2. Blow up P along S 0 to obtain P0 . Let E denote the resulting exceptional divisor and E10 , ..., Eδ0 the proper transforms of the first group of exceptional divisors. Each Ei0 is isomorphic to Ei ' P3 blown up along two skew lines Q0i , Q00i ⊂ Ei . 3. Each Ei0 is a P1 -bundle over P1 × P1 . Indeed, through each p ∈ Ei not on Q0i and Q00i passes a unique line l intersecting Q0i and Q00i . The bundle map is given by p 7→ (l ∩ Q0i , l ∩ Q00i ) ∈ Q0i × Q00i . Blow down each Ei0 to P1 × P1 . The resulting X is isomorphic to BlS P4 . Remark 1.3. The blowup X → P4 has a quadric surface Qi , i = 1, ...δ, over each node of S . Then P0 is obtained as the blowup of X along these quadrics. Let π1 : X → P4 be the blowup along S and π2 : X → P4 the resolution of f so that 9 π2 = π1 ◦ f . We organize these maps into a diagram: P0 ~ P X π1 π2   f  P4 / P4 Note that, by the definition of blowup, X is exactly the graph of f . Let L (resp. M) denote the divisor of the hyperplane class of the left (resp. right) P4 . We also use L (resp. M) to denote its pullbacks to X, P and P0 (resp. X and P0 ). It’s clear that L4 = 1. We have (1.1) M4 = 1 on X as f is birational. We define n by (1.2) L3 M = n and ξ by (1.3) LM 3 = ξ. We may interpret n and ξ as the degrees of the homogeneous forms inducing f and f −1 respectively. Define m as the multiplicity of S in the base locus. It is clear that M = nL − m E + 2Σδi=1 Ei0   on P0 . 10 Since a nondegenerate subvariety in projective space has degree greater than one, the linear system in |M| inducing P0 → P4 must be complete. Thus we have h0 P0 , M = 5.  (1.4) We use these equations in our classification of Cremona transformations in Section 1.4. 1.1.2 Computing the intersection numbers Let Σ denote the normalization of S and let KΣ be its canonical class. Then the blowup of Σ along the preimage of the nodes is isomorphic to S 0 . We denote by C a general sectional curve of S and also its preimages in Σ and S 0 . Let d = C 2 = deg S . Note that Ei ∩ S 0 = Q0i ∪ Q00i are exactly the exceptional curves on S 0 over the i-th node. Lemma 1.4. We have LEi0 = 0. We also have E 3 Ei0 = −4, E 2 Ei02 = 2, EEi03 = 0 and Ei04 = −1. Proof. First, LEi0 = 0 since their intersection is empty. Recall that Ei0 is isomorphic to Ei ' P3 blown up at skew lines Q0i and Q00i . Write D E Pic(E 0 ) = H, Q i e0 , Q e00 where H is the polarization from P3 while Q e0 and Q e00 are the excep- e00 = 0 and Q e0 Q tional divisors over the lines. We clearly have Q e0 H 2 = Q e00 H 2 = 0. Since NQ0i /P3 = OQ0i (1) ⊕ OQ0i (1) then writing ζ = c1 (OP(NQ0 /P3 ) (1)) we obtain ζ 2 + 2Hζ = 0 in the i Chow group of Q = P(NQ0i /P3 ). We have Q |Qe0 = −ζ so that e0 e0 e02 H = −ζH = −1, Q e03 = ζ 2 = −2Hζ = −2. Q 11 We have NEi0 /P0 = O(−H) and E|Ei0 = Q e0 + Q e00 . Thus we obtain E 3 Ei0 = (Q e0 + Q e00 )3 = Q e03 + Q e003 = −4 E 2 Ei02 = (Q e0 + Q e00 )2 (−H) = 2 EEi03 = (Q e0 + Q e00 )(−H)2 = 0 Ei04 = (−H)3 = −1.  Lemma 1.5. The intersection numbers involving L and E are 1. L3 E = 0 and L2 E 2 = −d 2. LE 3 = −5d − KΣC 3. E 4 = −15d − 5KΣC − c2 (Σ) + 6δ 4. E 4 = d2 − 25d − 10KΣC − KΣ 2 + 4δ Proof. L3 E = 0 since a general line doesn’t intersect S . We have L2 E 2 = − deg S 0 = −d. Assume that C = S ∩ L for some hyperplane L ' P3 . Then LE 3 = s(C, L)0 the zeroth Segre class of C in L, which equals [c(NC/L )−1 ]0 = [c(C)c(ι∗ T P3 )−1 ]0 = [([C] − KΣC − C 2 )([C] − 4d)]0 = −5d − KΣC. We have E 4 = −s(S 0 , P)0 = −[c(NS 0 /P )−1 ]0 = −[c(S 0 ) c(P)|−1 S 0 ]0 . Let  : P → P be the 4 blowup. The blowup formula for Chern classes gives c(P) =  ∗ c(P4 ) + (1 + Σi Ei )(1 − Σi Ei )4 − 1 = ([P] + L)5 + Σi (−3Ei + 2Ei 2 + 2Ei 3 − 3Ei 4 ). 12 Thus we have i + Qi )] c(P)|S 0 = ([S 0 ] + 5C + 10C 2 ) + Σi [−3(Q0i + Q00i ) + 2(Q02 002 = [S 0 ] + 5C − 3Σi (Q0i + Q00i ) + 10d − 4δ and also S 0 = [S ] − 5C + 3Σi (Qi + Qi ) + 15d − 14δ. c(P)|−1 0 0 00 Let τ : S 0 → Σ be the blowup. Then we have c(S 0 ) = [S 0 ] − τ∗ KΣ − Σi (Q0i + Q00i ) + c2 (Σ) + 2δ. Multiply the results to get E 4 = −15d − 5KΣC − c2 (Σ) + 6δ. Another expression for E 4 is derived from −[c(NS 0 /P )−1 ]0 = c2 (NS 0 /P ) − c1 (NS 0 /P )2 . We have c2 (NS 0 /P ) = d2 − 4δ. On the other hand c1 (NS 0 /P ) = c1 (T P )|S 0 − c1 (T S 0 ) = − (−5L + 3Σi Ei )|S 0 − (−KS 0 ) = 5C + τ∗ KΣ − 2Σi (Q0i + Q00i ), hence we deduce = 25d + 10KΣC + KΣ 2 − 8δ 2 c1 NS 0 /P0 and also     E4 = d2 − 4δ − 25d + 10KΣC + KΣ 2 − 8δ = d2 − 25d − 10KΣC − KΣ 2 + 4δ. 13  1.2 K3 surfaces of degree 12 and the construction In this section, we first recall some basic facts on K3 surfaces, then use Mukai’s construc- tion [Muk88] to produce an explicit example of a degree-12 K3 surface R ⊂ P7 together with three points p1 , p2 , p3 ∈ R. This example helps us prove the following theorem: Theorem 1.6. Let R ⊂ P7 be a generic K3 surface of degree 12 and Π := {p1 , p2 , p3 } ⊂ R a generic triple of points. 1. projection from Π maps R to a surface S ⊂ P4 with three nodes; 2. the complete linear system M of quartics vanishing along S cuts out S scheme- theoretically; 3. M induces a birational map f : P4 d P4 ; 4. the base locus of the inverse f −1 is also a projection of a degree-12 K3 surface Rˆ ⊂ P7 from three points. 1.2.1 Derived equivalences of K3 surfaces A (polarized) K3 surface is a smooth compact complex surface R with both the canonical bundle and the first Betti number being trivial, equipped with a polarization H, that is, an ample divisor which is indivisible in the Picard group Pic(R). Its degree is defined as the self-intersection d = H 2 . The moduli space Md of degree d K3 surfaces is nonempty of dimension 19 for each even d ≥ 2. Two smooth projective varieties are derived equivalent if their bounded derived cate- gories of coherent sheaves are isomorphic. For K3 surfaces this equivalence relation can 14 be translated into an isomorphism of Hodge structures. The Mukai lattice of a K3 surface R is the sum ˜ Z) = H 0 (R, Z) ⊕ H 2 (R, Z) ⊕ H 4 (R, Z) H(R, equipped with a weight-two Hodge structure which extends the standard one on H 2 (R, Z) by taking H 0 (R, Z) and H 4 (R, Z) as (1, 1)-classes. The intersection pairing on this lattice is defined by (r1 , D1 , s1 ) · (r2 , D2 , s2 ) = D1 · D2 − r1 · s1 − r2 · s2 . Each coherent sheaf E yields a Mukai vector v(E) = (r(E), c1 (E), s(E)), where r(E) is the rank and r(E) + s(E) = χ(E). Mukai [Muk87] has shown that the second cohomology of a moduli space Mv (R) may be expressed   ⊥  v if v · v ≥ 2    H (Mv (R), Z) =  2   v⊥ /Zv if v · v = 0     provided v is primitive and satisfies certain technical conditions. A derived equivalence between R and Rˆ induces an isomorphism between Hodge structures Φ : H(R, e Z) ∼ / H( ˆ Z) e R, which may be chosen so that Rˆ = Mv (R) with Φ(v) = (0, 0, 1). This further induces an ˇ where T (R) := NS(R)⊥ ⊂ isomorphism between the transcendental lattices T (R) and T (R), H 2 (R, Z). 15 Theorem 1.7 ([Muk87, Orl97]). Let R and Rˆ be K3 surfaces. Then the following condi- tions are equivalent 1. R and Rˆ are derived equivalent. 2. H(R, e Z) and H( ˆ Z) are Hodge isometric. e R, ˆ are Hodge isometric. 3. T (R) and T (R) Theorem 1.8 ([HLOY03]). Suppose that R has Picard rank one and degree 2n. Let τ(n) be the number of prime factors of n. Then the number of isomorphism classes of K3 surfaces derived equivalent to R is equal to 2τ(n)−1 . 1.2.2 K3 surfaces of degree 12 A general degree-12 K3 surface R admits a unique partner Rˆ nontrivially derived equivalent to it. Let H and Hˆ denote the polarizations on R and Rˆ respectively. We may interpret Rˆ as a moduli space of vector bundles on R and vice versa [Muk99]. Let M(2,H,3) (R) denote the moduli space of rank-two stable bundles E with c1 (E) = H and χ(R, E) = 5, which is isomorphic to R. ˆ The universal bundle E → R × Rˆ induces a Hodge isometry Φ : H(R, e Z) ∼ / H( ˆ Z) e R, described previously. We have Φ(2, H, 3) = (0, 0, 1), Φ(0, 0, 1) = (2, H, ˆ 3) and Φ restricts to the isogeny on transcendental cohomology. It follows formally that Φ(1, 0, −2) = (−1, 0, 2), 16 thus after a shift the Mukai vector of ideal sheaves of length-three subschemes of R goes ˆ We obtain an isomorphism to the Mukai vector of length-three subschemes of R. H 2 (R[3] , Z) ' H 2 (Rˆ [3] , Z) of Hodge structure arising from Mukai lattices. Thus the Torelli Theorem [Mar11, Cor. 9.9] yields a birational equivalence R[3] ∼ / Rˆ [3] . A geometric construction of this map is given in Corollary 1.11. Remark 1.9. We also have Φ(1, 0, −1) = (1, H, ˆ 5). Elements of M(1,H,5) ˆ (R)ˆ may be interpreted as IZ (H) ˆ where Z ⊂ Rˆ has length two. Similar ∼ reasoning gives R[2] d Rˆ [2] . 1.2.3 Linear sections on the spinor tenfold Let V  C10 be a vector space equipped with a nondegenerate bilinear form q. The maxi- mal isotropic subspaces form two 10-dimensional subvarieties S+ , S− ⊂ Gr(5, V) isomorphic to each other, called orthogonal Grassmannians and are denoted by OG(5, V). Mukai [Muk88] proves that a general member of M12 is obtained as a linear section of OG(5, V) in a unique way modulo the action of the spin group Spin(10). ˆ of derived equivalent K3 surfaces of degree 12 is via One way to construct a pair (R, R) classical projective duality. Fix a maximal isotropic subspace U ⊂ V. Its exterior algebra 17 ∧• U decomposes into the even and odd parts ∧• U = ∧even U ⊕ ∧odd U naturally dual to each other. The group Spin(10) acts on the two spaces via the half-spin representation and its dual, and the orthogonal Grassmannians are realized as the orbits S+ ⊂ P15 , S− ⊂ Pˇ 15 ˇ are obtained as which are projectively dual to each other. Then the pair (R, R) R = S+ ∩ P, Rˆ = S− ∩ Pˇ where P ⊂ P15 and Pˇ ⊂ Pˇ 15 are dual 7-planes. The second interpretation allows us to write down the equations explicitly. Let U ∗ be the orthogonal complement of U with respect to q. Then OG(5, V) can be identified scheme theoretically as the zero locus in 2 ^ 4 ^ P(C ⊕ U⊕ U) ' P15 of the quadratic form V2 V4 det U ⊕ U ⊕ U ∗ −→ U ⊕U (1.5) (x, Ω, v) 7−→ (x(v) + 21 Ω ∧ Ω, Ω(v)) 18 (See, for example, [IM04, §2].) Here the isomorphism C ' det U is chosen and is com- patible with 4 U ' U ∗ . This gives us ten quadrics: V x0 x11 + x5 x10 − x6 x9 + x7 x8 −x1 x12 + x2 x13 − x3 x14 + x4 x15 x0 x12 + x2 x10 − x3 x9 + x4 x8 x1 x11 − x5 x13 + x6 x14 − x7 x15 (1.6) x0 x13 + x1 x10 − x3 x7 + x4 x6 −x2 x11 + x5 x12 − x8 x14 + x9 x15 x0 x14 + x1 x9 − x2 x7 + x4 x5 x3 x11 − x6 x12 + x8 x13 − x10 x15 x0 x15 + x1 x8 − x2 x6 + x3 x5 −x4 x11 + x7 x12 − x9 x13 + x10 x14 . 1.2.4 The data for our example The proof of Theorem 1.6 requires the computation on an explicit example via computer algebra system. Let z = (z0 , ..., z7 ) be homogeneous coordinates for P7 . We define a linear embedding ι : P7 ,→ P15 by x=z·H where    −1 3 2 0 2 −3 −1 0 3 1 0 3 0 2 0 −3    1 0 −3 0 −2 1 1 0 −2 −1 −1 −1 0 4 0 2        −1 −3 −2 0 −3 0 3 2 −1 −3 −1 2 −1 2 0 3     3 0 0 2 2 3 0 1 2 −1 0 2 −1 −2 2 3  H =   .   0 −1 1 −1 0 1 −3 3 2 2 1 3 0 −3 0 −3    1 0 0 0 0 −1 0 1 0 0 0 0 0 0 0 0        2 1 0 1 0 0 0 1 0 0 1 0 0 0 0 0    3 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 Our K3 surface is obtained as the section R = ι−1 (OG(5, V)). The last three rows of H are chosen as solutions of (1.6) so that they form a triple of points Π = {p1 , p2 , p3 } ⊂ R. Under 19 this choice the projection from Π is defined as π: P7 d P4 (z0 , ..., z7 ) 7→ (z0 , ..., z4 ). We set S = π(R). We manipulate this example by Singular [DGPS15] over the finite field †7 . We com- pute that S is singular along three nodes {a1 , a2 , a3 } and is the base locus of a Cremona transformation f : P4 d P4 . Moreover, the base locus of the inverse ( f )−1 is again a surface T singular along three nodes {b1 , b2 , b3 }. The matrix H is chosen such that the preimage of {a1 , a2 , a3 } on R and the preimage of {b1 , b2 , b3 } on the normalization of T are †7 -rational points. This is the smallest field where our computer could quickly find such an H. 1.2.5 Proof of Theorem 1.6 First, we prove that Theorem 1.6 holds for our example. We verify the following properties by computer over †7 : 1. S is singular along three points. They are nodal since the preimage of each singular point on R has two points outside Π. 2. The ideal of S is generated by five quartics f 0 , ..., f 4 . The double-point formula [Ful98, Thm. 9.3] indicates that the three nodes of (1) exist over characteristic zero. Indeed, let  : Σ → S be the normalization. Then Σ is isomorphic to R blown up at three points. The double-point class „() ∈ CH0 (Σ) is given by the 20 formula „() =  ∗ ∗ [Σ] − [ ∗ c(P4 ) · c(Σ)−1 ]0 = hS , S iP4 − ( ∗ c2 (P4 ) −  ∗ c1 (P4 ) · c1 (Σ) − c1 (Σ)2 + c2 (Σ)). It’s easy to verify that „() = 6. The quantity 12 „() = 3 counts the number of singularities on S with multiplicity if the singular locus is a finite set. Therefore (1) implies that the singular locus of S consists of three nodes. This proves Theorem 1.6(1). The five quartics f 0 , ..., f 4 lift to a basis f0 , ..., f4 for the ideal of S over characteristic zero. In particular Theorem 1.6(2) holds. The forms f0 , ..., f4 define a rational map f := ( f0 , ..., f4 ) : P4 d P4 which reduces to f := ( f 0 , ..., f 4 ) : P4 d P4 over †7 . The degree of f is computed by the self-intersection M 4 , which can be expanded as the right-hand side of equation (1.24). It’s easy to check that our example satisfies (n, m, d, δ) = (4, 1, 9, 3), KΣC = 3 and c2 (Σ) = 27. Inserting these data into (1.24) we get M 4 = 1, i.e. the map f is birational. Thus Theo- rem 1.6(3) holds. The inverse ( f )−1 can be calculated by computer. It consists of five quartics also and the base locus is a surface T singular along three points. These are nodes since each point has two preimage points on the normalization. By the same reasons as above, the base locus of f −1 is again a surface cut by five quartics and singular along three nodes. Then 21 Theorem 1.6(4) follows from Theorem 1.19. Now we prove Theorem 1.6 in the generic case. It’s clear that (1), (2) and (3) of the theorem are open conditions, so they hold for a generic example. As a consequence of Theorem 1.19, property (4) holds once Bs( f −1 ) is a surface cut out by five quartics and singular along three nodes. These are open conditions again so Theorem 1.6 holds for a generic example. 1.3 Derived equivalence via Cremona transformation Theorem 1.6 introduces a pair (RL , R M ) of degree-12 K3 surfaces, their images S L and S M in P4 , and the commutative diagram X π1 π2 ~ f RL / SL ⊂ P4 / P4 ⊃ SM o RM where BlS L P4 ' graph( f ) = X = graph( f −1 ) ' BlS M P4 Theorem 1.10. The two K3 surfaces RL and R M are derived equivalent. They are non- isomorphic if they have Picard number one. Corollary 1.11. The construction gives a birational map σ : R[3] [3] L d R M between the Hilbert schemes of length three subschemes. Our general approach is to prove that T (RL ) is isometric to T (R M ) by showing that both of them can be identified as the transcendental sublattice of H 4 (X, Z). Then we show that the induced isomorphism on the discriminant groups is nontrivial, which implies that RL and R M are not isomorphic to each other. 22 The locus contracted by the map f and the relation in the Grothendieck ring induced by the construction will be discussed in the end of the section. 1.3.1 The middle cohomology of X Retain the notation of Section 1.1. Let HL be the polarization of RL . Let F1 , F2 and F3 be the exceptional curves from the projection RL d S L . We consider HL , F1 , F2 and F3 as curves on S L . Their strict transforms H eL , F e1 , F e3 on X together with L2 and the quadrics e2 , F Q1 , Q2 , Q3 form a rank 8 sublattice AL (X) ⊂ H 4 (X, Z). We have eL2 = −HL2 = −12, H ei2 = −Fi2 = 1 F and Q2i = −Ei04 = 1 where i = 1, 2, 3. These classes are mutually disjoint, so the intersection matrix for AL (X) is L2 H eL F e1,2,3 Q1,2,3 L2 1 (1.7) H eL −12 F e1,2,3 I3×3 Q1,2,3 I3×3 where I3×3 is the identity matrix of rank 3. Lemma 1.12. There is a decomposition H 4 (X, Z) ' H 4 (X, Z)alg ⊕⊥ T (RL )(−1). 23 where H 4 (X, Z)alg is the sublattice spanned by algebraic classes. We have H 4 (X, Z)alg = AL (X) when RL has Picard number one. Here we use Λ(−1) to denote a lattice Λ equipped with the negative of its original product. Proof. We apply the blowup formula for cohomology to the composition P0 → P → P4 and the map P0 → X to obtain two decompositions for H 4 (P0 , Z). Then we compare them to get our result. Let S 0L ⊂ P be the strict transform of S L . Recall that S 0L is isomorphic to RL blown up at 3+6=9 points, where 3 are from the projection RL d S L while 6 are from the resolution S 0L → S L . Thus we have H 2 (S 0L , Z) ' Fi , Q0i , Q00i i=1,2,3 ⊕ H 2 (RL , Z). e0 and Q Let Q e00 be the strict transforms of Q0 and Q00 on P0 . Since P0 → P is the blowup i i i i along S 0L , we have H 4 (P0 , Z) ' H 4 (P, Z) ⊕ H 2 (S 0L , Z)(−1) (1.8) D E ' L2 , Ei02 , F ei , Q e0 , Q i e00 i ⊕ H 2 (RL , Z)(−1). i=1,2,3 For every i, we have i = −Qi = 1, i = −Qi = 1, e02 Q 02 e002 Q 002 e0i = Ei02 Q Ei02 Q e00i = 0 24 and Ei04 = −1. With these it’s straightforward to prove the isometry D E D E Ei02 , Q e0i , Q e0i , Ei02 + Q e00i ' Ei02 + Q e00i , Ei02 + Q e0i + Q e00i , whence (1.8) equals D E H 4 (P0 , Z) ' i i e00 , E 02 + Q e0 , E 02 + Q Ei02 + Q i i i e00 , L2 , F e0 + Q i ei i=1,2,3 (1.9) ⊕H 2 (RL , Z)(−1). By the description of the map Ei d Qi , the two fiber classes on Qi ' P1 × P1 pullback to hyperplanes in Ei containing either Q0i or Q00i , which correspond to the classes −Ei02 − Q e0 i e00 on P0 , respectively. The map P0 → X is the blowup along Qi , i = 1, 2, 3, so or −Ei02 − Q i D E (1.10) H 4 (P0 , Z) ' Ei02 + Q e0i , Ei02 + Q e00i ⊕ H 4 (X, Z) i=1,2,3 Combining (1.9) and (1.10) we get D E⊥ H 4 (X, Z) ' Ei02 + Q e0 , E 02 + Q i i e00 i i=1,2,3 D E ' Ei02 + Q e0 + Q i e00 , L2 , F i ei ⊕ H 2 (RL , Z)(−1). i=1,2,3 Both Qi and Ei02 + Q e0 + Q i ei=1,2,3 and H 2 (RL , Z), and Q2 = e00 are orthogonal to L2 , F i i (Ei02 + Q e0 + Q i e00 )2 = 1, so Qi = ±(E 02 + Q i i e0 + Q i e00 ). Therefore i D E H 4 (X, Z) ' Qi , L 2 , F ei ⊕ H 2 (RL , Z)(−1) i=1,2,3 D E ' Qi , L 2 , F ei ⊕ NS (RL )(−1) ⊕⊥ T (RL )(−1) i=1,2,3 ' H 4 (X, Z)alg ⊕⊥ T (RL )(−1) where NS (RL ) is the N´eron-Severi lattice of RL . 25 D E When RL has Picard number one, we have NS (RL )(−1) ' H eL . In this case D E H 4 (X, Z)alg ' Qi , L2 , F ei , H eL = AL (X). i=1,2,3  Lemma 1.12 also proves the decomposition H 4 (X, Z) ' H 4 (X, Z)alg ⊕⊥ T (R M )(−1) from the side of f −1 . So there is an isometry T (RL ) ' H 4 (X, Z)⊥alg (−1) ' T (R M ) which allows us to conclude that Proposition 1.13. RL and R M are derived equivalent. 1.3.2 The discriminant groups For an arbitrary lattice Λ with dual lattice Λ∗ := Hom(Λ, Z), we denote by dΛ := Λ∗ /Λ its discriminant group. Let A M (X) be the lattice constructed in the same way as AL (X) from the side of f −1 . Assume RL and R M have Picard number one. Then Lemma 1.12 implies that there is an isometry ∼ ϕ : A M (X) ⊕⊥ T (R M )(−1) → − AL (X) ⊕⊥ T (RL )(−1) such that ϕ = ϕA ⊕ ϕT with respect to the decompositions. It induces the commutative 26 diagram ϕA∗ dA M (X) / dAL (X) ∼ ∼ ∼  ϕT ∗  dT (R M ) / dT (RL ). ∼ These groups are all isomorphic to Z/12Z. From the intersection matrix (1.7) we know eL /12. Similarly, dA M (X) is generated by −H that dAL (X) is generated by −H eM /12 where H M is the polarization of R M and H eM is the strict transform on X. Lemma 1.14. We have the following equations in H 4 (X, Z) 1. M 2 = 7L2 − 3H eL + 4(F e1 + F e2 + F e3 ) + 2(Q1 + Q2 + Q3 ) eM = 36L2 − 17H 2. H eL + 24(F e1 + F e2 + F e3 ) + 12(Q1 + Q2 + Q3 ) Proof. The following computation is based on Corollary 1.16. Assume that M 2 = aL2 + bH eL + f1 F e1 + f2 F e2 + f3 F e3 + g1 Q1 + g2 Q2 + g3 Q3 . Then a = L2 M 2 = 7. For i = 1, 2, 3, we have gi = M 2 Qi = −M 2 Ei02 = −(4L − E − 2Σ j E 0j )2 Ei02 = −(−E − 2Σ j E 0j )2 Ei02 = −E 2 Ei02 − 4EEi03 − 4Ei04 = −2 − 0 + 4 = 2. e be the strict transform of the sectional curve C on P0 . Note that Let C LM = 4L2 − LE = 4L2 − C e = 4L2 − H eL + Σi F ei , 27 so we find 4 = LM 3 = (LM)M 2 = (4L2 − H eL + Σi F ei )(7L2 + bH eL + Σi f j F ej + 2Σk Qk ) = 28 + 12b + f1 + f2 + f3 and thus (1.11) f1 + f2 + f3 = −12b − 24. We also have 1 = M 4 = (7L2 + bH eL + Σi f j F ej + 2Σk Qk )2 = 49 − 12b2 + f12 + f22 + f32 + 12 which is equivalent to (1.12) f12 + f22 + f32 = 12b2 − 60. By the Cauchy-Schwarz inequality ( f1 + f2 + f3 )2 = ((1, 1, 1) · ( f1 , f2 , f3 ))2 (1.13) ≤ (1, 1, 1)2 ( f1 , f2 , f3 )2 = 3( f12 + f22 + f32 ). Applying (1.11) and (1.12) we get (−12b − 24)2 ≤ 3(12b2 − 60), i.e. 3b2 + 16b + 21 = (3b + 7)(b + 3) ≤ 0. The only integer solution is b = −3. Because (1.13) becomes an equality in this case, we have ( f1 , f2 , f3 ) = f (1, 1, 1) for some integer f . We obtain f = 4 by setting b = −3 in 28 (1.11). As a result, we find M 2 = 7L2 − 3H eL + 4(F e1 + F e2 + F e3 ) + 2(Q1 + Q2 + Q3 ). Next, assume that eM = aL2 + bH H eL + f1 F e1 + f2 F e2 + f3 F e3 + g1 Q1 + g2 Q2 + g3 Q3 . eL Ln M 2−n = H By symmetry, H eM M n L2−n for n = 0, 1, 2. In particular, a=H eM L2 = H eL M 2 = −3H eL2 = 36. We have 12 = H eL + Σi F eL (4L2 − H ei ) = H eL (LM) = H eM (ML) = (36L2 + bH eL + Σi fi F ei + Σ j g j Q j )(4L2 − H eL + Σi F ei ) = 144 + 12b + f1 + f2 + f3 . Rearrange to obtain (1.14) f1 + f2 + f3 = −12b − 132. Applying the symmetry again, we get 0=H eL L2 = H eM M 2 = (36L2 + bH eL + Σi fi F ei + Σ j g j Q j )(7L2 − 3H eL + 4Σi F ei + 2Σ j Q j ) = 252 + 36b + 4( f1 + f2 + f3 ) + 2(g1 + g2 + g3 ) whence 2( f1 + f2 + f3 ) + (g1 + g2 + g3 ) = −18b − 126 29 and combining with (1.14) gives (1.15) g1 + g2 + g3 = 6b + 138. We also have −12 = H L e2 = (36L2 + bH e2 = H M eL + Σi fi F ei + Σ j g j Q j )2 = 1296 − 12b2 + f12 + f22 + f32 + g21 + g22 + g23 , from which we obtain (1.16) f12 + f22 + f32 + g21 + g22 + g23 = 12b2 − 1308. By the Cauchy-Schwarz inequality, (−12b − 132)2 = ( f1 + f2 + f3 )2 = ((1, 1, 1) · ( f1 , f2 , f3 ))2 (1.17) ≤ (1, 1, 1)2 ( f1 , f2 , f3 )2 = 3( f12 + f22 + f32 ) and (6b + 138)2 = (g1 + g2 + g3 )2 = ((1, 1, 1) · (g1 , g2 , g3 ))2 (1.18) ≤ (1, 1, 1)2 (g1 , g2 , g3 )2 = 3(g21 + g22 + g23 ). Add the two inequalities and then apply (1.16) to get (−12b − 132)2 + (6b + 138)2 (1.19) ≤ 3( f12 + f22 + f32 + g21 + g22 + g23 ) = 3(12b2 − 1308) 30 which can be arranged as 2b2 + 67b + 561 = (2b + 33)(b + 17) ≤ 0. The only integer solution is b = −17 which makes (1.19) an equality. This forces (1.17) and (1.18) to be equalities also. Therefore ( f1 , f2 , f3 ) = f (1, 1, 1) and (g1 , g2 , g3 ) = g(1, 1, 1) for some integers f and g. We get f = 24 from (1.14) and g = 12 from (1.15). As a con- sequence, eM = 36L2 − 17H H eL + 24(F e1 + F e2 + F e3 ) + 12(Q1 + Q2 + Q3 ).  ∼ Proposition 1.15. The isomorphism ϕA ∗ : dA M (X) → − dAL (X) equals multiplication by 7 on Z/12Z. Proof. Recall that ϕA acts as the identity map on H 4 (X, Z)alg , thus ϕA (H eM ) = H eM . By Lemma 1.14 we have ϕA ( H eM ) = 36L2 − 17H eL + 24(F e1 + F e2 + F e3 ) + 12(Q1 + Q2 + Q3 ). as a map from A M (X) to AL (X). Therefore eM ) = −3L2 + ϕA ∗ (− 121 H 17 e H 12 L e1 + F − 2(F e2 + F e3 ) − (Q1 + Q2 + Q3 ) = −17 · (− 121 H eL ) mod AL (X) = 7 · (− 121 H eL ) mod AL (X).  Remark. By the symmetry the Cremona transformation f , the rank-8 lattice H 4 (X, Z)alg 31 is also spanned by the classes {M 2 , H eM , G e1 , G e2 , G e3 , K1 , K2 , K3 } e1 , G constructed in a similar way from the right-hand side. Here G e2 , G e3 are from the excep- tional curves and K1 , K2 , K3 are from the nodes. The full transformation between the two set of bases is       7 −3 4 4 4 2 2 2   M2 L2                     H eM     36 −17 24 24 24 12 12 12   H eL              G e1     4 −2 3 3 3 2 1 1   e1  F           G e2   4 −2 3 3 3 1 2 1   F e2   =   ·            G e3   4 −2 3 3 3 1 1 2   F e3               K1   2 −1 2 1 1 1 1 1   Q1              K2   2 −1 1 2 1 1 1 1   Q2          K3 Q       2 −1 1 1 2 1 1 1 3 This expression is unique up to the ordering of the exceptional curves and the nodes on each side. The top two rows are computed by Lemma 1.14. The other rows can be com- puted in a similar way. 1.3.3 Proofs of Theorem 1.10 and its Corollary We first prove the theorem. The derived equivalence follows from Proposition 1.13. Note that this implies that the Picard numbers of RL and R M are the same. Assume RL and R M have Picard number one. Suppose they are isomorphic. Then there 32 is an isometry θ : T (RL ) → ˜ T (R M ) which induces the isomorphism θ∗ : dT (RL ) → ˜ dT (R M ) − H12L 7→ − H12M e e under the identifications dT (RL ) ' dAL (X) and dT (R M ) ' dA M (X). By Proposition 1.15, the composition ϕA ◦ θ is an automorphism on T (RL ) acting as multiplication by 7 on dT (RL ). This contradicts the fact that the only automorphism on T (RL ) is the identity [Ogu02]. Hence RL and R M can’t be isomorphic to each other. Next we prove the corollary. The corollary is trivial if RL and R M are isomorphic, so we assume that they are non-isomorphic. Given a generic triple of points ΠL ∈ R[3] L , we determine a degree 12 K3 surface R M and a triple of points Π M ∈ R[3] M through the following steps: 1. Project RL from ΠL to obtain S L ⊂ P4 , whose ideal defines a Cremona transformation f : P4 d P4 . 2. The base locus of f −1 is a surface S M singular along three nodes. Normalize S M to get Σ M . 3. Σ M is the blowup of a degree 12 K3 surface R M along three points. The three excep- tional curves on Σ M are contracted to Π M ∈ R[3] M. Recall that a pair of derived equivalent K3 surfaces of degree 12 uniquely determines 33 each other up to isomorphism. So R M is independent of the choice of ΠL ∈ R[3] L by Theo- rem 1.10. Hence there is a rational map σ : R[3] L d R[3] M ΠL 7→ Π M . It is birational because ΠL is uniquely determined by Π M through the same process as above. 1.3.4 Locus contructed by the map Retain the notation of Section 1.1. Our example has d := deg(S L ) = 9, KΣC = 3, c2 (Σ) = 27, δ = 3 and M = 4L − E − 2Σ3i=1 Ei0 . Evaluating Lemmas 1.4 and 1.5 with this data yields Corollary 1.16. We have 1. LEi0 = 0, E 3 Ei0 = −4, E 2 Ei02 = 2, EEi03 = 0, Ei04 = −1, 2. L3 E = 0, L2 E 2 = −9, LE 3 = −48, E 4 = −159. Thus consequently, 3. L3 M = 4, L2 M 2 = 7, LM 3 = 4, M 4 = 1. Let X0 , ..., X4 be the homogeneous coordinates for P4 . The Cremona transformation f is ramified along the locus Θ where the Jacobian matrix ∂ fi ! Df = ∂X j 5×5 34 is degenerate. So Θ is a degree 15 hypersurface in P4 defined by det(D f ) = 0. This locus is called P-locus, which is classically defined as the image of the exceptional divisor of the blowup π2 [Dol12, §7.1.4]. In particular, Θ is irreducible. It also follows that Θ is the locus contracted by f and its image is the base locus S M . Proposition 1.17. The locus Θ ⊂ P4 contracted by f is an irreducible hypersurface of degree 15. It has multiplicity four along S L . Moreover, it equals the union of all of the 4-secant lines to S L . The analogous statement holds for the inverse f −1 by symmetry. Proof. Let m be the multiplicity of Θ along S L . Then the divisor class of its pullback to X equals ΘX = 15L − mE X . Here we use E X to denote the exceptional divisor of the blowup π1 . Because Θ is contracted onto a surface, we have 0 = M 3 ΘX = M 3 (15L − mE X ) = 60 − mM 3 E X . By definition, E X is mapped onto the P-locus of the inverse map f −1 . In particular, it is again a degree 15 hypersurface in P4 by symmetry. So M 3 E X = 15, which implies m = 4. Let F X be the exceptional locus of the blowup π2 . We have L = 4M − F X by symmetry, hence F X = 4M − L. (Note that this equals 15L − 4E X = ΘX ) The fiber of the map F X → S M over a smooth 35 point is represented by the class FX M2 1 lX = = FX M2. deg S M 9 The image l = π1 (lX ) is a rational curve of degree 1 1 1 L · lX = LF X M 2 = L(4M − L)M 2 = (16 − 7) = 1. 9 9 9 The intersection number between l and S L can be computed by E X · lX = 1 E F M2 9 X X = 19 (4L − M)(4M − L)M 2 = 1 9 (64 − 28 − 4 + 4) = 4. Hence the fibers of F X → S M away from the double points is mapped by π1 to 4-secant lines to S L . In other words, S L admits a family of 4-secant lines parametrized by the smooth locus of S M . Conversely, every 4-secant line l to S L satisfies l · M = l · (4L − E X ) = 4 − 4 = 0. So l is contracted to a point by f . Hence the union of the 4-secant lines to S L forms a 3-fold contained in Θ and thus coincides with Θ.  1.3.5 Zero divisors in the Grothendieck ring Let K0 (Var/C) denote the Grothendieck ring of complex algebraic varieties. It is the abelian group generated by isomorphism classes of complex algebraic varieties subject to 36 the relation [Z] = [U] + [Z − U] where U is an open subvariety of Z. The multiplication is induced by the Cartesian prod- uct: [X][Y] = [X × Y] which is associative and commutative with unit 1 = [Spec C]. More generally, if Z → X is a Zariski locally trivial bundle with fibers isomorphic to Y, by stratifying the base it’s easy to prove that [X][Y] = [Z]. Let Œ = [1 ] be the class of the affine line in K0 (Var/C). Consider a pair of non- isomorphic smooth projective varieties X and Y which are derived equivalent. It is inter- esting to know if there exists k ≥ 0 satisfying (1.20) ([X] − [Y])Œk = 0 and what the minimal k is if it exists [KS16]. When X is a generic K3 surface of degree 12, Ito, Miura, Okawa and Ueda [IMOU16] proves that there exists Y non-trivially derived equivalent to X such that (1.20) holds for k = 3. Actually, it can be improved to k = 1 straightforwardly from the point of view of the Cremona transformation. Theorem 1.18. Let RL and R M be a generic pair of K3 surfaces associated with our Cre- mona transformation. Then we have ([RL ] − [R M ])Œ = 0. 37 in K0 (Var/C). The relation is minimal in the sense that [RL ] − [R M ] , 0. Proof. Recall that ΣL is the normalization of S L as well as the blowup of RL at three points. Hence we have [S L ] = [ΣL ] − 3 = [RL ] + 3Œ − 3. From the blowup π1 : X → P4 we obtain [X] = ([P4 ] − [S L ]) + [π−1 1 (S L )] = ([P4 ] − [S L ]) + (([S L ] − 3)[P1 ] + [Q1 ] + [Q2 ] + [Q3 ]) = ([P4 ] − [S L ]) + (([S L ] − 3)[P1 ] + 3[P1 ]2 ) (1.21) = [P4 ] + 3[P1 ]([P1 ] − 1) + [S L ]([P1 ] − 1) = [P4 ] + 3[P1 ]Œ + [S L ]Œ = [P4 ] + 3[P1 ]Œ + [RL ]Œ + 3Œ2 − 3Œ. By symmetry, we also have (1.22) [X] = [P4 ] + 3[P1 ]Œ + [R M ]Œ + 3Œ2 − 3Œ. Subtracting (1.22) from (1.21) we get ([RL ] − [R M ])Œ = 0. Next we show that [RL ] , [R M ], and it is sufficient to show that [RL ] , [R M ] modulo Œ. According to [LL03], [RL ] = [R M ] mod Œ if and only if RL and R M are stably birational. Because a K3 surface is not rationally connected, this implies that RL and R M are birational and thus isomorphic, contradicting Theorem 1.10.  38 1.4 Exclusion of alternative constructions This section shows that there exists just one class of Cremona transformations of P4 that can be resolved by blowing up an irreducible surface S with nodes, i.e., the class con- structed in Section 1.2. Recall that [CK89] classified the case where S is smooth. Theorem 1.19. Let S ⊂ P4 be an irreducible surface of degree d with δ > 0 nodes. Assume there exists a Cremona transformation f : P4 d P4 resolved by blowing up S . Let n and ξ denote the degrees of the homogeneous forms inducing f and f −1 respectively, and m the multiplicity of S in the base locus. Then we have n = ξ = 4, m = 1, δ = 3, and S is obtained by projecting a degree 12 K3 surface from three points. The remainder of this section is devoted to the proof of Theorem 1.19. 1.4.1 Extracting Diophantine equations By Lemmas 1.4 and 1.5, equation (1.3) can be expressed as (1.23) ξ = n3 − 3nm2 d + m3 (KΣC + 5d) . Similarly, equation (1.1) can be expressed as (1.24) 1 = n4 − 6n2 m2 d + 4nm3 (KΣC + 5d) − m4 (15d + 5KΣC + c2 (Σ) − 6δ) 39 and equivalently as (1.24’) 1 = n4 − 6n2 m2 d + 4nm3 (KΣC + 5d) + m4 (d2 − 25d − 10KΣC − KΣ 2 + 4δ). The two formulas follow from the two expressions Lemma 1.5 (3) and (4) for E 4 , respec- tively. The right-hand sides of these equations are arranged as polynomials in n and m. Note that only the coefficients of m4 reflect the appearance of nodes. 1.4.2 Enumeration of combinatorial cases Lemma 1.20. Only the following (n, m, ξ) can occur. n m ξ (a) 3 1 2 (b) 4 1 4 (c) 7 2 3 (d) 9 2 9 (e) 43 10 7 (f) 24 5 24 (g) 49 10 49 Proof. In the smooth case, the same list [CK89, Theorem 1.6] is obtained by using [CK89, Lemma 0.2] and [CK89, Formulae 0.3]. The proof of the former proceeds unchanged even with the existence of the nodes. The latter can be derived from (1.23) and (1.24) and only the terms with power of m up to two matter, so the nodes don’t change the result. Therefore the same elimination process works and we obtain the same list.  40 1.4.3 Exclusion of cases Here we show that only Case (b) can occur. Lemma 1.21. Cases (c) and (e) do not occur. Proof. The proof is similar to the smooth case [CK89, Lemma 3.2]. Assume Case (c) holds. Then (1.23) reduces to 2KΣC = 11d − 85 and (1.24) reduces to 465 = 62d − 2c2 (Σ) + 12δ. This is odd on the left and even on the right, a contradiction. Assume Case (e) holds. Now (1.23) reduces to 79d = 795 + 10KΣC, so d is divisible by 5. On the other hand, (1.24) becomes −34188 = −11094d + 1720(KΣC + 5d) − 100(15d + 5KΣC + c2 (Σ) − 6δ). Note that 5 divides the right but not the left, a contradiction.  Lemma 1.22. Cases (d), (f) and (g) do not occur. Proof. Let IS be the ideal sheaf of S ⊂ P4 . Generally, the global sections of ISm (n) and OP0 (M) are bijective canonically. So we have (1.25) h0 (P4 , ISm (n)) = h0 (P0 , M) = 5 41 by equation (1.4). We prove the lemma case by case. In each case, we prove by contradiction in the following situations h0 (P4 , IS (4)) = 0, = 1 and ≥ 2. Assume Case (d) holds. Suppose h0 (P4 , IS (4)) = 0. Consider the surjective map M (1.26) H 0 (P4 , IS (k1 )) ⊗ H 0 (P4 , IS (k2 ))  H 0 (P4 , IS2 (9)). k1 +k2 =9 By hypothesis h0 (P4 , IS (k)) = 0 for all k ≤ 4. Since k1 + k2 = 9 implies k1 ≤ 4 or k2 ≤ 4, the left-hand side of (1.26) vanishes. Thus h0 (P4 , IS2 (9)) = 0, contradicting (1.25). Let X0 , ..., X4 be a basis of degree one forms on P4 in what follows. Suppose h0 (P4 , IS (4)) = 1. Let A ∈ H 0 (P4 , IS (4)) be a generator. This forces h0 (P4 , IS (k)) = 0 for all k ≤ 3. It follows that H 0 (P4 , IS2 (8)) is generated by A2 . Then (1.25) indicates that A2 X0 , ..., A2 X4 form a basis for H 0 (P4 , IS2 (9)). As a result, the linear system |IS2 (9)| defines an automorphism of P4 instead of a Cremona transformation. Suppose h0 (P4 , IS (4)) ≥ 2. Let A, B ∈ H 0 (P4 , IS (4)) be independent. Then A2 and AB are independent in H 0 (P4 , IS2 (8)). We claim that there exists an i such that A2 Xi is not a linear combination of ABX j , j = 0, ..., 4. Suppose not, i.e. A2 Xi = ABLi for some linear form Li , i = 0, ..., 4. Then we have A B = L0 X0 = L1 X1 , which implies that L0 = X0 L1 X1 , so X1 divides L1 . Therefore A B = L1 X1 is a scalar, thus A and B are dependent, a contradiction. As a result, there exists an i such that A2 Xi and ABX0 , ..., ABX4 form an independent subset of H 0 (P4 , IS2 (9)). Thus h0 (P4 , IS2 (9)) ≥ 6 > 5, a contradiction. Assume Case (f) holds. Suppose h0 (P4 , IS (4)) = 0. Then h0 (P4 , IS (k)) = 0 for all k ≤ 4. Now we consider the 42 map M (1.27) H 0 (P4 , IS (k1 )) ⊗ · · · ⊗ H 0 (P4 , IS (k5 ))  H 0 (P4 , IS5 (24)). k1 +···+k5 =24 At least one ki ≤ 4, i = 1, ..., 5, if their sum equals 24. Hence the left-hand side of (1.27) vanishes. Thus h0 (P4 , IS2 (24)) = 0 , 5. Suppose h0 (P4 , IS (4)) ≥ 1. Let A ∈ H 0 (P4 , IS (4)) be a nonzero element. Then A5 ∈ H 0 (P4 , IS5 (20)). Multiplication by A5 defines an injection ·A5 : H 0 (P4 , OP4 (4)) ,→ H 0 (P4 , IS5 (24)). 8 Thus h0 (P4 , IS5 (24)) ≥ 4 = 70 > 5, a contradiction. The elimination of Case (g) is similar to Case (f). In Case (g), we use the surjection M H 0 (P4 , IS (k1 )) ⊗ · · · ⊗ H 0 (P4 , IS (k10 ))  H 0 (P4 , IS10 (49)) k1 +···+k10 =49 to rule out the situation h0 (P4 , IS (4)) = 0. If H 0 (P4 , IS (4)) contains A , 0, then multipli-   cation of A10 with 9-forms produces 134 = 715 independent elements in H 0 (P4 , IS10 (49)), which is not allowed.  Lemma 1.23. In cases (a) and (b) we have 43 (a) (3, 1, 2) (b) (4, 1, 4) d ≤8 ≤ 15 KΣC 4d − 25 7d − 60 KΣ2 d2 − 11d + 4δ + 30 d2 + d + 4δ − 105 c2 (Σ) 19d − 95 + 6δ 46d − 405 + 6δ 12χ(OΣ ) d2 + 8d − 65 + 10δ d2 + 47d − 510 + 10δ 5d−23 g(C) 2 4d − 29 The invariants d and δ satisfy (d − 5)2 = 2δ in Case (a) and (d − 10)(d − 15) = 2δ in Case (b) respectively. Proof. In order to compute the invariants in the list, we first use (1.23) to express KΣC in d with given n, m and ξ. Then (1.24) (resp. (1.24’)) allows us to express c2 (Σ) (resp. KΣ2 ) in d and δ. We compute 12χ(OΣ ) and g(C) by Noether’s formula and the genus formula, respectively. The upper bound for d comes from the inequality d < (n/m)2 which holds generally [CK89, Formulae 0.3 (v)]. We have h0 (P4 , IS (n)) = h0 (P0 , M) = 5 by (1.4). On the other hand, h1 (P4 , IS (n)) = 0 by [Dol12, Prop. 7.1.4]. Hence h0 (P4 , IS (n)) = χ(P4 , IS (n)) = χ(P, IS 0 (n)) where the second equality follows from the functoriality of the Euler characteristic. The short exact sequence 0 → IS 0 (nL − 2Σi Ei ) → OP (nL − 2Σi Ei ) → OS 0 (nC − 2Σi (Q0i + Q00i )) → 0 44 implies that χ(P, IS 0 (n)) = χ(P, nL − 2Σi Ei ) − χ(S 0 , nC − 2Σi (Q0i + Q00i )). χ(P, nL − 2Σi Ei ) counts the dimension of the space of degree n polynomials singular along ∆, so n+4 ! χ(P, nL − 2Σi Ei ) = − 5δ. 4 By the previous computations and the Riemann-Roch formula, we have  1 12 (d2 − 10d + 385 − 62δ) for (a) χ(S , nC − 0 2Σi (Q0i + Q00i )) = 1 12 (d2 − 25d + 930 − 62δ) for (b) whence  − 1 (d2 − 10d − 35 − 2δ) for (a) 12 χ(P, IS 0 (n)) = − 121 (d2 − 25d + 90 − 2δ) for (b). Then the two equations are obtained by setting χ(P, IS 0 (n)) = 5.  Lemma 1.24. Case (a) does not occur. Proof. Assume (a) is satisfied. Then the same argument as in [CK89, Theorem 3.3] im- plies that d = 5. By Lemma 1.23 we have δ = 0.  1.4.4 Geometric analysis of the remaining case To complete the proof of Theorem 1.19, it remains to analyze the last possible case. 45 Lemma 1.25. We have (d, δ) = (8, 7) or (9, 3). The invariants in these cases are (d, δ) KΣC KΣ2 c2 (Σ) χ(OΣ ) g(C) (8, 7) −4 −5 5 0 3 (9, 3) 3 −3 27 2 7 Proof. By the previous part only Case (b) is allowed. By Lemma 1.23, we have d ≤ 15 and g(C) = 4d − 29 ≥ 0. Hence 8 ≤ d ≤ 15. Then (d − 10)(d − 15) = 2δ and our hypothesis δ > 0 force d = 8 or 9, which implies that δ = 7 or 3, respectively. The invariants are computed directly by using Lemma 1.23.  Consider the linear system |KΣ +C| for both cases of Lemma 1.25. We have h1 (KΣ +C) = 0 by Kodaira vanishing and h2 (KΣ + C) = h0 (−C) = 0 by Serre duality. Lemma 1.26. The case (d, δ) = (8, 7) is not allowed. Proof. By the Riemann-Roch formula, h0 (KΣ + C) = χ(KΣ + C) = χ(OΣ ) + 12 (KΣ + C)C = 0 + 12 (−4 + 8) = 2 Because c2 (Σ) = 5, Σ can’t be P2 , P1 × P1 or a minimal ruled surface. It implies that OΣ (KΣ + C) is generated by global sections [Som81, Prop. 2.2]. Hence the system |KΣ + C| defines a morphism φ : Σ → P1 , the adjunction mapping. 46 Consider the Stein factorization φ Σ / P1 > r s  Σ 0 where r is a proper morphism with connected fibers and s is a finite morphism. By [Som81, (2.3)], this leads to two possible situations: 1. dim φ(Σ) = 0. Here we have g(C) = 1, a contradiction. 2. dim φ(Σ) = 1. Then there exists a P1 -bundle π : R → Σ0 such that r factors as  / Σ R r   Σ0 , where Σ is the blowup of R in at most one point of each fiber blown up, and C meets the generic fiber with degree two. Furthermore, the map s is an isomorphism except possibly if g(C) = 3 and h1,0 (Σ) = 1. Let’s analyze Situation (2): The map s can’t be an isomorphism. Otherwise, R is a Hirzebruch surface and χ(OΣ ) = χ(OR ) = 1, a contradiction. Hence we obtain g(Σ0 ) = h1,0 (R) = h1,0 (Σ) = 1. Then χ(OΣ ) = 0 and c2 (Σ) = 5 implies that Σ has Hodge diamond 1 1 1 0 7 0. 47 Since the N´eron-Severi group of R has rank two, we conclude that Σ is the blowup of R along five points on distinct fibers, and R is ruled over the elliptic curve Σ0 . Let h be the class of a section on R and f be the class of a fiber so that h2 = m, h f = 1, and f 2 = 0 for some integer m. According to the description of (2), the image of C in R gives a class H = 2h + b f for some integer b and C =  ∗ H − 5i=1 Fi where F1 , ..., F5 are the exceptional P curves on Σ. Note that KΣ =  ∗ KR + 5i=1 Fi . Thus we have P 8 = C2 = H2 − 5 ⇒ H 2 = 13 −4 = KΣC = KR H + 5 ⇒ KR H = −9 and consequently χ(H) = χ(OR ) + 21 H(H − KR ) = 0 + 12 (13 + 9) = 11. On the other hand, one can use the exact sequence 0 → OR → OR (h) → Oh (m) → 0 to get χ(h) = m, and then use 0 → OR (h) → OR (2h) → Oh (2m) → 0 to obtain χ(2h) = 3m. Then an induction on n with the sequence 0 → OR (2h + (n − 1) f ) → OR (2h + n f ) → O f (2) → 0 48 implies that χ(H) = χ(2h + b f ) = 3m + 3b. But this implies 11 = χ(H) is divisible by 3, a contradiction.  Proposition 1.27. If S has a node, then it can only be the image of a K3 surface R ⊂ P7 of degree 12 projected from three points on R, and the number of nodes must be δ = 3. Proof. By the Riemann-Roch formula, h0 (KΣ + C) = χ(KΣ + C) = χ(OΣ ) + 12 (KΣ + C)C = 2 + 12 (3 + 9) = 8. Because c2 (Σ) = 27, Σ can’t be P2 , P1 × P1 or a minimal ruled surface. It follows that OΣ (KΣ + C) is generated by global sections [Som81, Prop. 2.2]. Hence |KΣ + C| defines an adjunction morphism with Stein factorization φ Σ / P7 > r s  Σ. 0 There are three possible situations [Som81, (2.3)]: 1. dim φ(Σ) = 0. We have g(C) = 1, a contradiction. 2. dim φ(Σ) = 1. Then r : Σ → Σ0 is again obtained by blowing up a P1 -bundle, with no more than one point in a fiber blown up. In particular, 1 ≥ χ(OΣ ) = 2, a contradiction. 3. dim φ(Σ) = 2. Then r : Σ → Σ0 expresses Σ as the blowup of a smooth surface Σ0 along a finite set with F · C = 1 for any positive dimensional fiber F of r. Moreover, s : Σ0 → P7 is an embedding. 49 Now we are in Situation (3). Let F1 , ..., Fk be the exceptional curves on Σ relative to r and let H be the very ample divisor on Σ0 which defines s. Then k X k X C=r H−∗ Fi and KΣ = r KΣ0 + ∗ Fi i=1 i=1 and it follows that 9 = C2 = H2 − k 3 = KΣC = KΣ0 H + k −3 = KΣ2 = KΣ20 − k. By the Riemann-Roch formula, 8 = χ(H) = χ(OΣ0 ) + 21 H(H − KΣ0 ) = 2 + 21 ((9 + k) − (3 − k)) = 5 + k, which implies that k = 3. Hence Σ is obtained by blowing up Σ0 along three distinct points, and Σ0 ⊂ P7 has deg(Σ0 ) = H 2 = 12, KΣ0 H = KΣ20 = 0, c2 (Σ0 ) = 24, χ(OΣ0 ) = 2. We claim that Σ0 is a K3 surface. Indeed, its Kodaira dimension κ , 2 since KΣ0 H = 0. If κ = 1, then Σ0 has minimal model R an elliptic surface, such that nKR is numeri- cally equivalent to a positive linear combination of some fiber classes if n is large enough [Bea96, Prop. IX.3]. This implies that KΣ0 is numerically effective which contradicts to the fact that KΣ0 H = 0. If κ = −∞, then h1,0 (Σ0 ) = 0 and thus 1 ≥ χ(OΣ0 ) = 2, a contradiction. As a result, Σ0 has κ = 0 and thus is a K3 surface. Besides, the birational map R d Σ → S can be realized as the projection from three 50 points on R. Furthermore, the fact that δ = 3 can also be verified directly by the double- point formula as in Section 1.2.5.  Chapter 2 Nodal rational scrolls and the Hilbert schemes One of the main methods in analyzing the rationality of a cubic fourfold is to make use of the algebraic surfaces inside it. There are certain cases where rational scrolls appear naturally in a cubic fourfold, such that the existence of a unirational parametrization for the cubic, or even the rationality, can be determined via this incidence relation. Rational scrolls in cubic fourfolds are usually singular. In order to exploit such scrolls, we need to gain a good understanding of their singularities and moduli spaces. In Section 2.1, we review the background and classify the singularities on rational scrolls. In Section 2.2, we provide a way to construct nodal rational scrolls in P5 , which allows us to analyze the hypersurfaces cutting through them with the aid of computer algebra systems. In Section 2.3, we describe a stratification of the corresponding Hilbert scheme and estimate the dimensions. Because the main purpose of this chapter is to develop the technical details required by Chapter 3, we suggest the reader to start with Chapter 3 and use this chapter as a reference. 51 52 2.1 Rational scrolls and the singularities 2.1.1 Hirzebruch surfaces Let E = OP1 ⊕ OP1 (m), m ≥ 0 be a locally free sheaf on P1 . We define the Hirzebruch surface Fm as the P1 -bundle Fm = P(E ) → P1 . Let f denote the fiber class, and let g = c1 (OP(E ) (1)) be the class of a section. Then the canonical divisor is KFm = −2g + (m − 2) f , and we have Pic(Fm )  Z f ⊕ Zg with intersection pairing given by f g f 0 1 g 1 m. Moreover, the divisor h = ag + b f is ample if and only if it is very ample if and only if a > 0 and b > 0 [Har77, Chapter V, §2.18]. Lemma 2.1. If h = ag + b f is ample, then hi (Fm , h) = 0 for i > 0 and am h0 (Fm , h) = (a + 1)( + b + 1). 2 These formulas appear in several places in the literature with slightly different details depending on the contexts. See, for example, [Laf02, Prop. 2.3], [BBF04, p.543], and [Cos06, Lemma 2.6]. They can be proved by induction on the integers a and b, or by applying the projection formula to the bundle map Fm → P1 . 53 The surface Fm admits a deformation to Fm−2k for all m > 2k ≥ 0. Here is an explicit example: n o F = x0 m y1 − x1 m y2 + tx0 m−k x1 k y0 = 0 ⊂ P1 × P2 × C, where ([x0 , x1 ], [y0 , y1 , y2 ], t) is the coordinate of P1 × P2 × C [BPVdV84, p. 205]. This is a family over C with (2.1) F0  Fm and Ft  Fm−2k for t , 0. A more conceptual way to describe the deformation goes as follows. The elements of Ext1 P1 (E, E) are in one-to-one correspondence with the deformations of E over the dual C[t] numbers Dt  (t2 ) [Har10, Th. 2.7]. More precisely, the element in Ext1 P1 (E, E)  Ext1 P1 (OP1 (m), OP1 ) represented by the exact sequence 0 → OP1 → OP1 (k) ⊕ OP1 (m − k) → OP1 (m) → 0, for some k satisfying m > 2k ≥ 0, corresponds to a coherent sheaf E on P1 × Dt , flat over Dt , such that E0  E and Et  OP1 (k) ⊕ OP1 (m − k) for t , 0. This induces a flat family F over Dt as in (2.1). Lemma 2.2. [Sei92, Lemma 1. and Theorem 4.] Let T Fm be the tangent bundle over Fm . Then H 2 (Fm , T Fm ) = 0, and there are natural isomorphisms H 1 (Fm , T Fm )  H 1 (P1 , OP1 (−m))  Ext1 P1 (E, E). 54 In particular, Fm admits an analytic versal deformation with a base manifold of dimension h1 (P1 , OP1 (−m)) = m − 1. 2.1.2 Rational normal scrolls Fisrt recall that, for n ≥ 1, a rational normal curve is a nondegenerate smooth curve C ⊂ Pn of degree n. It is projectively equivalent to the Veronese embedding P1 ,→ Pn via the linear series |OP1 (n)| [Har95, Example 1.14 & Prop. 18.9]. In particular, C admits the parametric equation C −→ Pn (2.2) s 7−→ (1, s, ..., sn ). Moreover, such curve is characterized by the property that every n distinct points on C are linearly independent. Let 1 ≤ u ≤ v < N be positive integers such that N = u + v + 1. Choose disjoint linear subspaces P1  Pu and P2  Pv in PN . Note that PN is spanned by P1 and P2 under the assumption. Choose two rational normal curves C 1 ⊂ P1 and C2 ⊂ P2 , and fix an isomorphism ϕ : C1 → C2 . Collecting the lines p ϕ(p) for p ∈ C1 , we obtain a smooth surface [ S u,v = p ϕ(p), p∈C1 called a rational normal scroll of type (u, v). We say S u,v is balanced if u = v or u + 1 = v. 55 The line p ϕ(p) is called a ruling of S u,v . When u < v, the curve C1 is called the directrix of S u,v . The above construction determines S u,v uniquely up to projective isomorphism. In particular, we can parametrize S u,v by C2 −→ PN (2.3) (s, t) 7−→ (1, s, ..., su , t, st, ..., sv t). Let (z0 , ..., zN ) be homogeneous coordinates for PN . Then the 2 × 2-minors of the matrix    z . . . zv−1 zv+1 . . . zN−1   0  .  (2.4)   z ... zv zv+2 . . . zN  1 cut out S u,v ⊂ PN [Har95, Lect. 9]. The rulings of S u,v form a rational curve in the Grassmannian G(1, N) of lines in PN . With (2.2), we can parametrize this curve by C −→ G(1, N)   (2.5)  1 s ... su 0 0 ... 0   ,   s 7−→   0 0 ... 0 1 s ... sv  where we represent an element in G(1, N) by a basis of its underlying line. Every nondegenerate irreducible surface of degree N − 1 in PN is either a rational normal scroll scroll or the Veronese surface [GH94, p. 522-525]. In particular, each S u,v ⊂ PN is an embedded Hirzebruch surface via (2.6) |g + u f | : Fv−u ,→ PN . 56 Note that, if S u,v is balanced, then it is the image of F0  P1 × P1 or F1  Blo P2 . Lemma 2.3. Every hyperplane section of S u,v not containing a ruling is a rational normal curve of degree u + v. Proof. Let H ⊂ PN be a hyperplane not containing a ruling of S u,v and set C = H ∩ S u,v . Then H intersects each ruling in a single point. This induces an isomorphism between C1 (or C2 ) and C via the φ, which implies that C is a smooth rational curve. Under the embedding (2.6), C ⊂ H  Pu+v has degree (g + u f )2 = u + v and the inclusion is defined by the linear series |OC (u + v)|. Hence C is a rational normal curve.  Lemma 2.4. For m > 2k ≥ 0, let F be an abstract deformation of Hirzebruch surfaces over C such that F0  Fm and Ft  Fm−2k for t , 0. Let 1 ≤ u ≤ v be integers satisfying v − u = m. Then F can be realized as an embedded deformation S over C in Pu+v+1 such that S0  S u,v and St  S u+k,v−k for t , 0. Proof. This follows from the fact that H 1 (Fu−v , T PN |Fu−v ) = 0 under the embedding (2.6) and [Har10, Remark 20.2.1].  2.1.3 Rational scrolls Definition 2.5. We call a surface S ⊂ PN a rational scroll (or a scroll) of type (u, m + u) if it is the image of a Hirzebruch surface Fm via a birational morphism defined by an 57 N-dimensional subsystem d ⊂ |g + u f | for some u > 0. A rational scroll S ⊂ PN of type (u, v) is either a rational normal scroll S u,v , or the projection image of S u,v ⊂ PD+1 from a (D − N)-plane disjoint from S u,v , where D = u + v = deg(S u,v ) = deg(S ). In the latter case, we call a line on S a ruling if it comes from a ruling on S u,v . Lemma 2.6. Let ι : Fm ,→ PN be an embedding defined by a subseries of |ag + b f | with a > 0 and b > 0. Let S denote the image and let NS /PN be the normal bundle of S in PN . Then hi (S , NS /PN ) = 0 for all i > 0, and am h0 (S , NS /PN ) = (N + 1)(a + 1)( + b + 1) − 7. 2 In particular, if S is smooth of degree D then this formula reduces to h0 (S , NS /PN ) = (N + 1)(D + 2) − 7. Proof. The short exact sequence (2.7) 0 → T S → T PN |S → NS /PN → 0 induces the long exact sequence 0 → H 0 (S , T S ) → H 0 (S , T PN |S ) → H 0 (S , NS /PN ) → H 1 (S , T S ) → H 1 (S , T PN |S ) → H 1 (S , NS /PN ) → H 2 (S , T S ) → H 2 (S , T PN |S ) → H 2 (S , NS /PN ) → 0. 58 In order to calculate the dimensions in the right, we need the dimensions in the first two columns. For the middle column, we restrict the Euler exact sequence 0 → OPN → OPN (1)⊕(N+1) → T PN → 0 to S and then obtain (2.8) 0 → OS → OS (h)⊕(N+1) → T PN |S → 0. Lemma 2.1 implies that hi (S , OS (h)) = 0 for i > 0. Hence the long exact sequence of (2.8) equals 0 → H 0 (S , OS ) → H 0 (S , OS (h))⊕(N+1) → H 0 (S , T PN |S ) → 0 with all the other terms vanishing. It follows that h0 (S , T PN |S ) = (N + 1)h0 (S , OS (h)) − h0 (S , OS ) = (N + 1)(a + 1)( am 2 + b + 1) − 1. For the first column, one can use the Hirzebruch-Riemann-Roch formula to compute that χ(S ) = 6. We also have h2 (S , T S ) = 0 by Lemma 2.2. Thus h0 (S , T S ) − h1 (S , T S ) = χ(S ) = 6. As a result, the long exact sequence for (2.7) takes the form 0 → H 0 (S , T S ) → H 0 (S , T PN |S ) → H 0 (S , NS /PN ) → H 1 (S , T S ) → 0 → H 1 (S , NS /PN ) → 0 → 0 → H 2 (S , NS /PN ) → 0. 59 Therefore hi (S , NS /PN ) = 0, ∀i > 0, and h0 (S , NS /PN ) = h0 (S , T PN |S ) − χ(S ) = (N + 1)(a + 1)( am 2 + b + 1) − 7. When S is a rational scroll, we have ag + b f = g + b f . Then the reduced form is obtained from a = 1 and D = (g + b f )2 = m + 2b.  2.1.4 Singularities on rational scrolls We assume that D = u + v ≥ N ≥ 5. The singular locus of a scroll S ⊂ PN is nonempty only if it is the image of a rational normal scroll S u,v ⊂ PD+1 via a nontrivial projection. Let q : S u,v → S be the projection map. It is easy to see that a point p ∈ S is singular if and only if either • p = q(l) ∩ q(l0 ) is isolated for two distinct rulings l, l0 ⊂ S u,v , or • p belongs to a singular line q(l) ⊂ S , which occurs when q is ramified at l ⊂ S u,v . Definition 2.7. Let S be a surface. We call a singular point p ∈ S a node if, locally around p, S has two smooth branches meeting transversally at p. Assume that the singular locus of S has dimension zero. Then each singular point p ∈ S is set-theoretically the intersection of two or more rulings, and p is a node when there are exactly two. Let m be the number of the rulings passing through a singular point.   Then the number of singularities on S is counted as m2 . Let Sec(S u,v ) be the secant variety and let T (S u,v ) ⊂ Sec(S u,v ) be the tangent variety. Then every x ∈ Sec(S u,v ) − T (S u,v ) satisfies either 1. x lies on one and only one secant line, or 60 2. x lies on more than one secant lines. Let Z2 ⊂ Sec(S u,v ) denote the union of such points. Lemma 2.8. Z2 , ∅ if and only if u = 2. When this happens, Z2  P2 and each x ∈ Z2 lies on infinitely many secant lines. Proof. Let x ∈ Z2 be any point. Then there are two distinct secant lines α and β passing through x. Since S u,v is cut out by quadrics, every secant line intersects S u,v transversally in exactly two points. Let α ∩ S u,v = {a1 , a2 } and β ∩ S u,v = {b1 , b2 }. We claim that a1 , a2 , b1 , b2 line on four distinct rulings of S u,v . If they line on two rulings l1 , l2 ⊂ S u,v , then (2.9) span(l1 , l2 ) = span(a1 , a2 , b1 , b2 ) = span(α, β)  P2 . This is impossible since two distinct rulings are linearly independent [Har95, Exercise 8.21]. If they lie on three rulings, then the projection from P1 πP1 : PN d P2 contracts span(α, β) to a line trisecant to C2 . But C2 has no trisecant line since every v points on C2 are linearly independent, and v ≥ 3 under our assumption. Hence the claim holds. Assume that a1 , a2 , b1 , b2 lie on the rulings l1 , l2 , l3 , l4 , respectively. We can find a hyperplane H which intersects l1 , l2 , l3 , l4 in a1 , a2 , b1 , b2 transversally. Then the irreducible component C ⊂ H ∩ S u,v passing through a1 , a2 , b1 , b2 is a rational normal curve with 61 deg(C) ≥ u ≥ 1 [Har95, Example 8.17]. The linear dependence (2.9) forces C to be either a line or a conic. If C is a line then C = l1 = l2 , a contradiction. Hence C is a conic.. It follows that u ≤ deg C = 2. If u = 1, then C dominates C2 via πP1 . But this cannot happen since C2 has degree v ≥ 4 in this condition. Hence u = 2 and thus v ≥ 3 > u. This implies that C coincides with the directrix . It follows that the Z2 is the 2-plane spanned by C and each point of Z2 lies on infinitely many secants. Conversely, u = 2 implies that Z2 contains the 2-plane spanned by C2 . By the same arguement as above they coincide and every x ∈ Z2 lies on infinitely many secants.  Corollary 2.9. Assume that S is the projection of S u,v from a (D − N)-plane Q ⊂ PD+1 . The scroll S is singular along r points if and only if Q intersects Sec(S u,v ) in r points away from T (S u,v ) ∪ Z2 . Proof. Assume that S has isolated singularities. The number r counts the number of the pair (l, l0 ) of distinct rulings on S u,v such that q(l) intersects q(l0 ) in one point. (Different pairs might intersect in the same point.) Thus it counts the number of the unique line joining l, l0 and Q. By Lemma 2.8, each x ∈ Sec(S u,v ) away from T (S u,v ) ∪ Z2 lies on a unique secant. Thus it is the same as the number of the intersection between Q and Sec(S u,v ) away from T (S u,v ) ∪ Z2 .  Proposition 2.10. Retain the notation in Cor. 2.9 Assume that u = 1 and that Q∩T (S 1,v ) = ∅. Then S has isolated singularities. Proof. If Q intersects Sec(S 1,v ) in points, then the proposition follows from Corollary 2.9. So we assume that Q ∩ Sec(S 1,v ) contains a curve Γ. We claim that Γ ∩ T (S u,v ) , ∅ under the assumption, which then contradicts to our hypothesis. Let f be the fiber class of S 1,v . Then the linear system | f | parametrizes the rulings of S 1,v . For distinct l, l0 ∈ | f |, the linear span of l and l0 is a 3-space Pl,l0 ⊂ Sec(S 1,v ). 62 Consider the incidence correspondence “ = {(x, l + l0 ) ∈ PD+1 × |2 f | : x ∈ Pl,l0 }. Observe that “ is a P3 -bundle over |2 f |  P2 via the second projection p2 : “ → |2 f |. On the other hand, the image of “ under the first projection p1 : “ → PD+1 is the secant variety Sec(S 1,v ). Consider the diagonal ∆ := {2l : l ∈ | f |} ⊂ |2 f |. It’s easy to see that the tangent variety T (S 1,v ) ⊂ Sec(S 1,v ) is the image of p−1 2 (∆) via the first projection. If Γ 1 Pl,l0 for all l + l0 , then the curve p−1 1 (Γ) is mapped to a curve in |2 f | which intersects ∆ nontrivially. It follows that Γ ∩ T (S u,v ) , ∅. Assume that Γ ⊂ Pl,l0 for some l + l0 . The directrix C1 is a line in Pl,l0 by hypothesis, so we have T (S u,v ) ∩ Pl,l0 = P ∪ P0 where P and P0 are the 2-planes spanned by C1 and l and l0 , respectively. So Γ and T (S u,v ) has a nontrivial intersection in Pl,l0 .  63 2.2 Constructing nodal scrolls in P5 We introduce a method to construct rational scrolls in P5 of type (1, v) with isolated singu- larities. During the construction we will show that Proposition 2.11. Assume v ≥ 4. There exists a rational scroll in P5 of type (1, v) with isolated singularities which has at least r singularities if there are four odd integers a ≥ b ≥ c ≥ d > 0 satisfying 1. 8r + 4 = a2 + b2 + c2 + d2 , 2. a + b + c ≤ 2v − 3. The construction allows us to produce an explicit example which can be manipulated by a computer algebra system. We verify that Proposition 2.12. There exists a degree-9 rational scroll S ⊂ P5 which has eight isolated singularities and smooth otherwise such that 1. h0 (P5 , IS (3)) = 6, where IS is the ideal sheaf of S in P5 . 2. S is contained in a smooth cubic fourfold X. 3. S deforms in X to the first order as a two dimensional family. 4. S is also contained in a singular cubic fourfold Y. Proposition 2.12 is going to be used in Section 3.3, Chapter 3. Fix a rational normal scroll S 1,v ⊂ PD+1 . Recall that, every degree D scroll S ⊂ P5 of type (1, v) is projectively equivalent to the projection of S 1,v from P⊥ for some P ∈ G(5, D + 1). 64 2.2.1 Plane k-chains Let k be a positive integer. It can be proved by induction that k distinct lines in a projective  space intersect in at most 2k points counted with multiplicity, and the maximal number is attained exactly when the k lines span a 2-plane. Definition 2.13. Let k ≥ 1 be an integer. We call the union of k distinct lines which span a 2-plane a plane k-chain. Let W ⊂ PN be the union of a finite number of lines. A plane k-chain in W is called maximal if it is not a subset of a plane k0 -chain in W for some k0 > k. Let S ⊂ P5 be a singular scroll with isolated singularities. There’s a subset W ⊂ S consisting of a finite number of rulings defined by [ (2.10) W= l : l is a ruling passing through a singular point on S . By Zorn’s lemma, W can be expressed as n [ W= Ki : Ki is a maximal plane ki -chain with ki ≥ 2. i=1 If two plane k-chains share more than one line, then they must lie on the same 2-plane. In particular, both of them can not be maximal. Therefore, for distinct maximal plane k- chains Ki and K j in W, we have either Ki ∩ K j = ∅ or Ki ∩ K j = l a single ruling. It follows P    that the number of singularities on S equals ni=1 k2i since a plane k-chain contributes 2k singularities. Let l1 , ..., lk ⊂ S 1,v be k distinct rulings which span a subspace Pl1 ,...,lk ⊂ PD+1 . The images of the rulings form a plane k-chain on S through projection if and only if Pl1 ,...,lk is projected onto a 2-plane in P5 . Parametrize the rulings as in (2.5) with l j = l j (s j ), j = 1, ..., k. Then Pl1 ,...,lk is spanned by the row vectors of the following (k + 2) × (D + 2) 65 matrix    1 s  1 0 0 ... 0    0 0 ... 0   1 s   2  P(s1 , ..., sk ) =  0 0 1 s1 ... s1  .   (2.11) v  ..   .       1 sk ... sk  v   0 0 The projection S 1,v → S is restricted from a linear map Λ : PD+1 d P5 . Suppose Λ is represented by a (D + 2) × 6 matrix   Λ= v1 v2 v3 v4 v5 v6 , where v1 , ..., v6 are vectors in PD+1 . Then Pl1 ,...,lk is projected onto a 2-plane if and only if the (k + 2) × 6 matrix P(s1 , ..., sk ) · Λ has rank three. 2.2.2 Control the number of nodes Let r be a non-negative integer. We introduce a method to find a projection Λ which maps S 1,v to a singular scroll S with isolated singularities. The method allows us to control the number of singularities such that it is bounded below by r. For simplicity, we consider only the cases when the configuration W ⊂ S defined in (2.10) consists of four disjoint 66 maximal plane k-chains. We start by picking distinct rulings on S u,v and produce four matrices P1 , P2 , P3 , and   P4 as in (2.11). Suppose Pi consists of ki rulings. Note that Pi contribute k2i singularities if its rulings are mapped to a plane ki -chain. Thus we also assume that ! ! ! ! k1 k2 k3 k4 (2.12) r= + + + . 2 2 2 2 Here we allow ki = 1 which means that Pi consists of a single ruling and thus contributes no singularity.   Consider Λ = v1 v2 v3 v4 v5 v6 as an unknown. Let P be the 5-plane spanned by v1 , ..., v6 . We are going to construct Λ satisfying 1. rk (Pi · Λ) = 3, i = 1, 2, 3, 4. 2. P⊥ ∩ T (S 1,v ) = ∅. Note that (1) makes the number of isolated singularities ≥ r, while (2) confirms that no curve singularity occurs. We divide the construction into two steps: Step 1. Find v1 , v2 , v3 and v4 to satisfy (1). Consider each Pi as a linear map by multiplication from the left. We are trying to find independent vectors v1 , v2 , v3 and v4 such that for each Pi three of them are in the kernel while the remaining one isn’t. The four vectors arranged in this way contribute exactly one rank to each Pi · Λ. In the next step, v5 and v6 will be general vectors in PD+1 satisfying some open conditions. This contributes two additional ranks to each Pi · Λ, which makes (1) true. Under the standard parametrization for S 1,v ⊂ PD+1 , the underlying vector space of PD+1 can be decomposed as A ⊕ B with A representing the first 2 coordinates and B repre- senting the last v + 1 coordinates. With this decomposition, the matrix P in (2.11) can be 67 decomposed into two Vandermonde matrices    1 s1 ... s1    v   1 s  ..  1   P =  A PB =  .  .    and   1 s   2  1 sk ... svk   Note that ker P = ker PB . So we can search for the vectors from ker PB . In our situation, we have four matrices P1 B , P2 B , P3 B , P4 B which have four kernels ker P1 B , ker P2 B , ker P3 B , and ker P4 B , respectively. By the assumption ki ≤ v and the fact that a Vandermonde matrix has full rank, each ker Pi B is a codimension ki subspace of B. Now we want to pick v1 , ..., v4 from B such that each ker Pi B contains exactly three of the four vectors, i.e. we want (2.13) ker Pi B ∩ {v1 , v2 , v3 , v4 } = 3, for i = 1, 2, 3, 4. One way to satisfy (2.13) is to pick vi from   \  (2.14)  ker P j B  − ker Pi B , for i = 1, 2, 3, 4.   j,i The sets in (2.14) are nonempty if and only if   dim ker Pα B ∩ ker Pβ B ∩ ker Pγ B ≥ 1 for all distinct α, β, γ ∈ {1, 2, 3, 4}. This is equivalent to (2.15) kα + kβ + kγ ≤ v, for distinct α, β, γ ∈ {1, 2, 3, 4}. 68 So we have to include (2.15) as one of our assumptions. Step 2. Adjust v1 , ..., v4 and then pick v5 and v6 to satisfy (2). Lemma 2.14. Let vi ⊥ be the hyperplane in PD+1 orthogonal to vi . The four vectors v1 , ..., v4 can be chosen generally such that 4i=1 vi ⊥ intersects T (S 1,v ) only in the directrix of S 1,v . T Proof. Parametrize the rational normal curve C = S 1,v ∩ P(B) by θ(s) = (0, 0, 1, s, ..., sv ). Then the standard parametrization (2.3) can be written as (1, s, 0, ..., 0) + tθ(s). Let a and b be the parameters for the tangent plane over each point. Then the tangent variety T (S 1,v ) has the parametric equation h i (1, s, 0, ..., 0) + tθ(s) + a (0, 1, 0, ..., 0) + t dθ ds (s) + bθ(s) = (1, s + a, 0, ..., 0) + (t + b)θ(s) + ta dθ ds (s). T4 ⊥ Each point on T (S 1,v ) lying in i=1 vi is a common zero of the equations ! dθ (2.16) (t + b) (θ(s) · vi ) + ta (s) · vi = 0, i = 1, 2, 3, 4. ds By considering (t + b) and ta as variables, (2.16) becomes a system of linear equations given by the matrix    θ(s) · v θ(s) · v θ(s) · v θ(s) · v  1 2 3 4  .     θ0 (s) · v θ0 (s) · v θ0 (s) · v θ0 (s) · v  1 2 3 4 69 The matrix fails to be of full rank exactly when s admits the existence of α, β ∈ C, αβ , 0, such that αθ(s) + βθ0 (s) · vi = 0, i = 1, 2, 3, 4.  (2.17) T4 ⊥ Note that (2.17) has a solution if and only if i=1 vi and the tangent variety T (C) of C intersect each other. One can choose v2 , v3 and v4 in general from (2.14) so that 4i=2 vi ⊥ is disjoint from C. T This forces 4i=2 vi ⊥ to intersect T (C) in either empty set or points. By the properties of T a rational normal curve, the hyperplane orthogonal to a point on C contains no invariant subspace when one perturb the point. Hence, after necessary perturbation of the chosen T4  rulings, one can choose v1 from (2.14) such that i=1 vi ⊥ ∩ T (C) = ∅. As a result, the equations in (2.16) become independent, so the solutions are t = b = 0 or a = 0, t = −b. Both solutions form the directrix of S 1,v .  With the above adjustment, we can pick v5 and v6 in general in PD+1 so that the (D − 5)- plane Q = v1 ⊥ ∩ ... ∩ v6 ⊥ has no intersection with T (S 1,v ). Note that the projection defined by Λ is the same as the projection from Q. By Proposition 2.10, this projection produces a rational scroll with isolated singularities. Proposition 2.15. There exists a rational scroll in P5 of type (1, v) with isolated singular- ities which has at least r singularities if there are four positive integers k1 ≥ k2 ≥ k3 ≥ k4 satisfying (2.12) and (2.15): ! ! ! ! k1 k2 k2 k2 r= + + + and k1 + k2 + k3 ≤ v. 2 2 2 2 Proposition 2.11 is obtained by expanding the binomial coefficients followed by a change of variables. 70 2.2.3 Proof of Proposition 2.12 In the following we exhibit an explicit example which can be manipulated by a computer algebra system over characteristic zero. The main program used in our work is Singular [DGPS15]. Consider P10 with homogeneous coordinate x = (x0 , ..., x10 ). We define the rational normal scroll S 1,8 by the 2 × 2 minors of the matrix    x x x x x x x x x   0 2 3 4 5 6 7 8 9  .    x x x x x x x x x  1 3 4 5 6 7 8 9 10 In order to project S 1,8 onto a rational scroll whose singular locus is zero dimensional and consists of at least eight singular points, we use the method introduced previously to construct a projection  T  T  v   1   0 0  0 120 −34 −203 91 70 −56 13 −1       0 0 2880 5184 −2372 −2196 633   v   2   261 −63 −9 2       v   0 0 0 480 304 −510 −339 30 36 0 −1    3  Λ =   =      v4   0 0 0 144 36 −196 −49 56 14 −4 −1          1 0   v5   1 1 1 1 1 1 1 1 1          v6 0 1 1 0 0 0 0 0 0 0 0 Let z = (z0 , ..., z5 ) be the coordinate for P5 . Then the projection P10 d P5 defined by Λ can be explicite written by z = x · Λ. Let S be the image of S 1,8 under the projection. Due to the limit of the author’s computer, we check that S has eight singularities and smooth otherwise over the finite 71 field of order 31. On the other hand, the double point formula implies that S has eight double points if the singular locus is isolated. Hence the singularity of S consists of eight double points over characteristic zero as required. The generators of the ideal of S contain six cubics, so property (1) is confirmed. Prop- erties (2) and (4) can be easily checked by examining the linear combinations of those cubics. The final step is to varify property (3). Let X ⊂ P5 be a smooth cubic containing S . Let F1 (S ) and F1 (X) denote the Fano variety of lines on S and X, respectively. Then it is equivalent to show that F1 (S ) deforms in F1 (X) to the first order with dimension two. Let G(1, 5) be the grassmannian of lines in P5 . Every element b ∈ G(1, 5) is parametrized by a 2 × 6 matrix      b   b  1   10 b11 b12 b13 b14 b15  =  (2.18)    b   b 20 b21 b22 b23 b24 b25  2 where b1 and b2 are two vectors which span the line b. Let PX = PX (z) be the homogeneous polynomial defining X. Let V be the 6-dimensional linear space underlying P5 . Consider PX as a symmetric function defined on V ⊕ V ⊕ V. Then F1 (X) ⊂ G(1, 5) is cut out by the four equations (2.19) PX (b1 , b1 , b1 ), PX (b1 , b1 , b2 ), PX (b1 , b2 , b2 ), PX (b2 , b2 , b2 ). Consider the Fano variety of lines on S 1,8 as a rational curve P1 ⊂ G(1, 10) parametrized by    r s 0 0 0 0 0 0 0 0 0  Q =      0 0 r8 r7 s r6 s2 r5 s3 r4 s4 r3 s5 r2 s6 rs7 s8  72 where (r, s) is the homogeneous coordinate for P1 . Then F1 (S ) ⊂ G(1, 5) is defined by the parametric equation R = Q · Λ. Now consider a 2 × 6 matrix dR whose first row consists of arbitrary linear forms on P1 while the second row consists of arbitrary 8-forms. The coefficients of those forms introduce 2 · 6 + 9 · 6 = 66 variables c1 , ..., c66 . Then an abstract first order deformation of F1 (S ) in G(1, 5) is given by R + dR. Inserting R + dR into (2.19) gives us four polynomials in r and s with coefficients in c1 , ..., c66 . The linear parts of the coefficients form a system of linear equations in c1 , ..., c66 whose associated matrix has rank 53. Then the first order deformation of F1 (S ) in F1 (X) appears as solutions of the system. In addition to the 53 constraints contributed by the above linear equations, we also have • 4 constraints from the GL(2) action on the coordinates (2.18). • 3 constraints from the automorphism group of P1 . • 4 constraints from rescaling the four equations (2.19). So F1 (S ) deforms in F1 (X) to the first order with dimension 66 − 53 − 4 − 3 − 4 = 2. 73 2.3 Hilbert schemes of rational scrolls Let N ≥ 3 be an integer. The Hilbert polynomial PS for a degree D smooth surface S ⊂ PN has the following form ! 1 2 1 PS (x) = Dx + D + 1 − π x + 1 + pa , 2 2 where π is the genus of a generic hyperplane section and pa is the arithmetic genus of S [Har77, V, Exercise 1.2]. We are interested in the case when S is a rational scroll. In this case π = pa = 0, so D 2 D  PS (x) = x + + 1 x + 1. 2 2 Every smooth surface sharing the same Hilbert polynomial has π = 0 and pa = 0 also and thus is rational. We denote by HilbPS (PN ) the Hilbert scheme of subschemes in PN with Hilbert polynomial PS . The closure of the locus parametrizing degree D scrolls forms a component HD ⊂ HilbPS (PN ). We study this space by stratifying it according to the types of the scrolls. Recall that, by fixing a rational normal scroll S u,v ⊂ PD+1 where D = u + v, a rational scroll S ⊂ PN of type (u, v) is either S u,v itself or the image of S u,v projected from a disjoint (D − N)-plane. We define Hu,v ⊂ HD as the closure of the subset consisting of smooth rational scrolls of type (u, v). In this section, we will first show that Proposition 2.16. Assume D + 1 ≥ N ≥ 3. 1. HD is generically smooth of dimension (N + 1)(D + 2) − 7. 2. Hu,v is unirational of dimension (D + 2)N + 2u − 4 − δu,v , where δu,v is the Kronecker delta. We also have 74 3. Hu,v ⊂ Hu+k,v−k for 0 ≤ 2k < v − u, and Hb D2 c,d D2 e = HD . When D+1 = N, a generic element of Hu,v is projectively equivalent to a fixed rational normal scroll S u,v ⊂ PD+1 . In this case Hu,v is birational to PGL(D + 2) quotient by the stablizer of S u,v . When D ≥ N, a generic element in Hu,v is the projection of S u,v from a (D − N)-plane. Note that Hu,v also records the scrolls equipped with embedded points along their singular loci. Such element occurs when the (D − N)-plane contacts the secant variety of S u,v . We r denote by Hu,v ⊂ Hu,v the closure of the subset parametrizing the schemes such that the singular locus of each of the underlying varieties consists of ≥ r isolated singularities. Let HDr ⊂ HD denote the union of Hu,v r through all possible types. The main goal of this section is to prove the following theorem Theorem 2.17. Assume D ≥ N ≥ 5, and assume the existence of a degree D rational scroll with isolated singularities in PN which has at least r singularities. Suppose rN ≤ (D + 2)2 − 1, then HDr has codimension at most r(N − 4) in HD . Especially when r = 1, HD1 is unirational of codimension exactly N − 4. 2.3.1 Parametrizing the Hilbert schemes Here we give a general picture of the component HD and also prove Proposition 2.16. Note that Proposition 2.16 (1) follows immediately from Lemma 2.6. As mentioned before, Hu,v is birational to PGL(D + 2) when D + 1 = N. In order to study the case of D ≥ N, we introduce the projective Stiefel variety. Definition 2.18. Let VN+1 (CD+2 ) = GL(D + 2)/GL(D − N + 1) be the homogeneous space of (N + 1)-frames in CD+2 . The group C∗ acts on VN+1 (CD+2 ) by rescaling, which induces a geometric quotient –(N, D + 1) that we call a projective Stiefel variety. 75 –(N, D + 1) has a fiber structure over G(N, D + 1): PGL(N + 1) ,→ –(N, D + 1) ↓ p G(N, D + 1) . An element Λ ∈ –(N, D + 1) over P ∈ G(N, D + 1) can be expressed as a (D + 2) × (N + 1)- matrix   Λ= v1 v2 . . . vN+1 (D+2)×(N+1) up to rescaling, where v1 , ..., vN+1 are column vectors which form a basis of the underlying vector space of P. In particular, each Λ ∈ –(N, D + 1) naturally defines a projection ·Λ : PD+1 d PN by multiplying the coordinates from the right. Let S u,v ⊂ PD+1 be the rational normal scroll given by the standard parametrization (2.3). When D ≥ N, every rational scroll in Hu,v is the image of S u,v under the projection defined by some Λ ∈ –(N, D + 1). So there is a dominant rational map π = π(S u,v ) : –(N, D + 1) d Hu,v (2.20) Λ 7−→ S u,v · Λ, where S u,v · Λ is the rational scroll given by the parametric equation C2 −→ PN (s, t) 7−→ (1, s, ..., su , t, st, ..., sv t) · Λ. Proof of Proposition 2.16 (2). Both PGL(D+2) and –(N, D+1) are rational quasi-projective varieties, so Hu,v is unirational either when D + 1 = N or D ≥ N by the above construction. The formula for the dimension of Hu,v holds by [Cos06, Lemma 2.6].  76 Proof of Proposition 2.16 (3). By Lemma 2.4, there exists an embedded deformation S in PD+1 over the dual numbers Dt = C[t] (t2 ) with S0  S u,v and St  S u+k,v−k for t , 0. For every rational scroll S ∈ Hu,v , we can find a Λ ∈ –(N, D + 1) such that S = S u,v · Λ. Then S · Λ defines an infinitesimal deformation of S to a rational scroll of type (u + k, v − k), which forces the inclusion Hu,v ⊂ Hu+k,v−k to hold. When (u, v) = (b D2 c, d D2 e), i.e. when u = v or u = v − 1, we have dim HD = dim Hu,v = (N + 1)(D + 2) − 7 by Proposition 2.16 (1) and (2). Because HD = u+v=D Hu,v , we must S have Hb D2 c,d D2 e = HD .  2.3.2 Projections that produce one singularity We are ready to study the locus in HD which parametrizes singular scrolls. Assume D ≥ N ≥ 5. Let us start from studying the projections that produce one singularity. Notations & Facts. Let K and L be any linear subspaces of PD+1 . 1. We use the same symbol to denote a projective space and its underlying vector space. The dimension always means the projective dimension. 2. Assume K ⊂ L, we write K ⊥L for the orthogonal complement of K in L. When L = PD+1 , we write K ⊥ instead of K ⊥P D+1 . 3. K + L means the space spanned by K and L. We write it as K ⊕ L if K ∩ L = {0}, and write it as K ⊕⊥ L if K and L are orthogonal to each other. The following two relations can be derived by linear algebra. (2.21) (K ∩ L)⊥ = K ⊥ + L⊥ . 77 (2.22) (K ∩ L)⊥K = (K ∩ L)⊥ ∩ K. Definition 2.19. Let l and l0 be a pair of distinct rulings on S u,v , and let Pl, l0 be the 3-plane spanned by them. We define σ(l, l0 ) to be a subvariety of G(N, D + 1) by n o σ(l, l0 ) = P ∈ G(N, D + 1) : dim(P ∩ P⊥l, l0 ) ≥ N − 3 . Lemma 2.20. Let p : –(N, D + 1) → G(N, D + 1) be the bundle map. Then p−1 ( σ(l, l0 ) ) ⊂ –(N, D + 1) consists of the projections which produce singularities by making l and l0 intersect. Proof. Let P ∈ G(N, D + 1) and Λ ∈ p−1 (P) be arbitrary. The target space of the projection map ·Λ is actually P. Let L ⊂ PD+1 be any linear subspace, then the image L · Λ is identical to (P⊥ + L) ∩ P. On the other hand, (2.21) and (2.22) implies that (P ∩ L⊥ )⊥P = (P ∩ L⊥ )⊥ ∩ P = (P⊥ + L) ∩ P. Therefore, N − 1 = dim P − 1 = dim(P ∩ L⊥ ) + dim(P ∩ L⊥ )⊥P = dim(P ∩ L⊥ ) + dim (P⊥ + L) ∩ P = dim(P ∩ L⊥ ) + dim (L · Λ) .  With L = Pl, l0 , the equation implies that dim(P ∩ P⊥l, l0 ) ≥ N − 3 ⇔ dim Pl, l0 · Λ ≤ 2.  It follows that p−1 ( σ(l, l0 ) ) = Λ ∈ –(N, D + 1) : dim(Pl, l0 · Λ) ≤ 2 .  78 The image Pl, l0 · Λ ⊂ PN lies in a plane if and only if l and l0 intersect each other after the projection ·Λ : PD+1 d PN . As a consequence, every Λ ∈ p−1 ( σ(l, l0 ) ) defines a projection which produces a singularity by making l and l0 intersect.  2.3.3 The geometry of the variety σ(l, l0 ) The properties of the singular scroll locus that we are interested in are the unirationality and the dimension. As a preliminary, we describe here the geometry of the variety σ(l, l0 ), which implies immediately the rationality of σ(l, l0 ) and also allows us to find its dimension easily. Instead of studying σ(l, l0 ) alone, the geometry would be more apparent if we consider generally the linear subspaces in PD+1 which satisfies a certain intersectional condition. Fix a (D − 3)-plane L ⊂ PD+1 . For every j ≥ 0, we define (2.23) σ j (L) = { P ∈ G(N, D + 1) : dim(P ∩ L) ≥ N − 4 + j } . For example, σ0 (L) = G(N, D + 1), and σ1 (P⊥l, l0 ) = σ(l, l0 ). Note that P ⊂ L or L ⊂ P if j ≥ min (4 , D − N + 1) in (2.23), so we have σ j (L) ) σ j+1 (L) if 0 ≤ j < min (4 , D − N + 1) , σ j (L) = σ j+1 (L) if j ≥ min (4 , D − N + 1) . Define σ◦j (L) = {P ∈ G(N, D + 1) : dim(P ∩ L) = N − 4 + j}, then σ◦j (L) = σ j (L) − σ j+1 (L) if 0 ≤ j < min (4 , D − N + 1) , σ◦j (L) = σ j (L) if j = min (4 , D − N + 1) . Lemma 2.21. Assume 1 ≤ j < min (4 , D − N + 1), then σ j (L) is singular along σ j+1 (L) 79 and smooth otherwise. The singularity can be resolved by a G(3 − j , D − N + 4 − j)-bundle over G(N − 4 + j , D − 3). Especially, σ j (L) is rational with codimension j(N − 3 + j) in G(N, D + 1). Proof. We define G j (L) to be the fiber bundle G(3 − j , D − N + 4 − j) ,→ G j (L) ↓ G(N − 4 + j , L) by taking G(3 − j , Q⊥ ) as the fiber over Q ∈ G(N − 4 + j , L). Apparently G j (L) is smooth and rational. We denote an element of G j (L) as (Q, R), where Q belongs to the base and R belongs to the fiber over Q. In the following, we will construct a birational morphism from G j (L) to σ j (L), which determines the rationality and the codimension immediately. Then we will study the sin- gular locus by analyzing the tangent cone to σ j (L) at a point on σ j+1 (L). Step 1. A birational morphism from G j (L) to σ j (L). Every P ∈ σ◦j (L) can be decomposed as P = (P ∩ L) ⊕⊥ (P ∩ L)⊥P . Because P ∩ L ∈ G(N − 4 + j , L) and (P ∩ L)⊥P is a (3 − j)-plane in (P ∩ L)⊥ , this induces a morphism ι : σ◦j (L) −→ G j (L)   P 7−→ P ∩ L , (P ∩ L)⊥P . On the other hand, Q ⊕⊥ R ∈ σ j (L) for every (Q, R) ∈ G j (L) since dim(Q ∩ L) = N − 4 + j by definition. Thus there is a morphism  : G j (L) −→ σ j (L) (2.24) (Q, R) 7−→ Q ⊕⊥ R. 80 Clearly, the composition  ◦ ι is the same as the inclusion σ◦j (L) ⊂ σ j (L). Therefore  is a birational morphism. The smoothness and rationality of G j (L) implies that σ◦j (L) is smooth and that σ j (L) is rational. Moreover, dim σ j (L) = dim G j (L) = (4 − j)(D − N + 1) + (N − 3 + j)(D − N + 1 − j) = (N + 1)(D − N + 1) − j(N − 3 + j) = dim G(N, D + 1) − j(N − 3 + j). Hence σ j (L) has codimension j(N − 3 + j) in G(N, D + 1). Step 2. The tangent cones to σ j (L). Choose any P ∈ σ j (L) and fix a φ ∈ T P G(N, D + 1)  Hom P, P⊥ . Let T P σ j (L) be  the tangent cone to σ j (L) at P. By definition, φ ∈ T P σ j (L) if and only if the condition dim(P ∩ L) ≥ N − 4 + j is kept when P moves infinitesimally in the direction of φ, which is equivalent to the condition that P ∩ L has a subspace Q of dimension N − 4 + j such that φ(Q) ⊂ L. Consider the decomposition ⊥ P⊥ = (P⊥ ∩ L) ⊕⊥ (P⊥ ∩ L)⊥P . Define ⊥   Γ : Hom P, P⊥ → Hom P ∩ L, (P⊥ ∩ L)⊥P  to be the composition of the restriction to P ∩ L followed by the right projection of the above decomposition. For any subspace Q ⊂ P ∩ L, φ(Q) ⊂ L if and only if φ(Q) ⊂ P⊥ ∩ L, if and only if 81 Q ⊂ ker Γ(φ). So L has a subspace Q of dimension N − 4 + j such that φ(Q) ⊂ L if and only if the (projective) dimension of ker Γ(φ) is at least N − 4 + j. Therefore, T P σ j (L) = φ ∈ Hom P, P⊥ : dim (ker Γ(φ)) ≥ N − 4 + j .   (2.25) Note that σ j (L) is the disjoint union of σ◦j+k (L) for all k satisfying 0 ≤ k ≤ min (4 , D − N + 1) − j. Assume P ∈ σ◦j+k (L), i.e. dim(P ∩ L) = N − 4 + j + k, then (2.25) is equivalent to T P σ j (L) = φ ∈ Hom P, P⊥ : rk Γ(φ) ≤ k .   (2.26) When k = 0, the constraint becomes rk Γ(φ) = 0, so T P σ j (L) = ker Γ is a vector space. This reflects the fact that σ j (L) is smooth on σ◦j (L) for all j. On the other hand, from the inequality dim(P ∩ L) + dim(P⊥ ∩ L) ≤ dim(L) − 1, we get dim(P⊥ ∩ L) ≤ dim(L) − dim(P ∩ L) − 1 = (D − 3) − (N − 4 + j + k) − 1 = D − N − j − k. It follows that   dim (P⊥ ∩ L)⊥P = dim(P⊥ ) − dim(P⊥ ∩ L) − 1 ⊥ ≥ (D − N) − (D − N − j − k) − 1 = j + k − 1.   So dim (P⊥ ∩ L)⊥P ≥ j + k − 1 ≥ k once k ≥ 1. Under this condition, the linear ⊥  ⊥  combination of members of rank k in Hom P ∩ L, (P⊥ ∩ L)⊥P can have rank exceeding k. So T P σ j (L) can not be a vector space, thus P is a singularity of σ j (L).  82 Recall that σ(l, l0 ) = σ1 (P⊥l, l0 ), so Lemma 2.21 implies that Corollary 2.22. σ(l, l0 ) is rational with codimension N − 2 in G(N, D + 1). 2.3.4 Families of the projections The singularities we have studied are those produced from the intersection of a fixed pair of distinct rulings. Now we are going to make use of the variety σ(l, l0 ) to control multiple singularities. [2] [2] Let P1 be the Hilbert scheme of two points on P1 and U ⊂ P1 be the open subset parametrizing reduced subschemes. On the rational normal scroll S u,v , the set of r pairs of distinct rulings (l1 + l1 0 , ..., lr + lr 0 ) : li , li 0 ∀i  is parametrized by U ×r . Let Σr be a subset of U ×r × G(N, D + 1) defined by  r   \  Σr =  + 0 , ..., + 0 , + σ(l , 0  .  ×r  (l l l l P) ∈ U × D 1) : P ∈ l )   1 1 r r G(N, i i     i=1 Let p1 be the left projection and p2 the right projection. Then there is a diagram p2 σ(li , li 0 ) Σr −→ G(N, D + 1) Tr i=1 ⊂ ↓ p1 ↓ (l1 + l1 0 , ..., lr + lr 0 ) ∈ U×r . By Lemma 2.20, the image p2 (Σr ) consists of the N-planes such that the projections to them produce at least r singularities. By the diagram above and Corollary 2.22, the codi- mension of Σr in U ×r × G(N, D + 1) is at most r(N − 4). When r = 1, Σ1 is rational with codimension exactly N − 4. 83 Our goal is to compute the dimension of p2 (Σr ), so we care about whether p2 is gener- ically finite onto its image or not. It turns out that the condition below is sufficient (See Lemma 2.24) There exists a rational scroll with isolated singularities (2.27) S ⊂ PN of type (u, v) which has at least r singularities. By considering S as the projection of S u,v from P⊥ for some N-plane P, we can apply Corollary 2.9 to translate (2.27) into the equivalent statement: There exists an N-plane P such that P⊥ intersects S (S u,v ) (2.27’) in ≥ r points away from T (S u,v ) ∪ Z2 . Proposition 2.23. (2.27) holds for r ≤ D − N + 1.    Proof. By [CJ96, Prop. 2.2] and [Har95, Example 19.10], deg S (S u,v ) = D−2 2 . Since dim S (S u,v ) = 5 and T (S u,v ) ∪ Z2 form a proper closed subvariety of S (S u,v ), we can    use Bertini’s theorem to choose a (D − 4)-plane R which intersects S (S u,v ) in D−2 2 points   outside T (S u,v ) ∪ Z2 . It is easy to check that D−2 2 ≥ D − N + 1. Thus we can choose D − N + 1 of the intersection points to span a (D − N)-plane Q ⊂ R. Then P = Q⊥ satisfies the hypothesis.  Unfortunately, Proposition 2.23 doesn’t cover the case D = 9, N = 5 and r = 8 in our proof of the unirationality of discriminant 42 cubic fourfolds. In the following, we estimate the dimension of p2 (Σr ) under the assumption (2.27) and leave the construction of examples to Section 2.2. Lemma 2.24. Suppose (2.27) holds. Then p2 (Σr ) has codimension ≤ r(N − 4) in G(N, D + 1). When r = 1, p2 (Σ1 ) is rational of codimension exactly N − 4. 84 Proof. Let l and l0 be distinct rulings on S u,v . We write Pl, l0 for the 3-plane spanned by them. Note that S (S u,v ) = l,l0 Pl, l0 . Let P be an N-plane satisfying (2.27’). Then there S exists r pairs of distinct rulings l1 , l1 0 , ..., lr , lr 0 such that P⊥ and Pli , li 0 intersect in exactly   one point for each pair li , li 0 . This implies that dim P⊥ + Pli , li 0 ≤ D − N + 3 for all i,     which is equivalent to dim P ∩ P⊥li , li 0 ≥ N − 3 for all i by (2.21). Hence P ∈ ri=1 σ li , li 0 , T  i.e. P belongs to the image of p2 . Suppose P⊥ intersects S (S u,v ) in m points, then the l1 + l1 0 , ..., lr + lr 0 in the preimage  of P is unique up to the choices of r from m pairs, the reordering of the r pairs, and the   transpositions of the rulings in a pair. Hence p2 is generically finite with deg p2 = mr · r!. In particular, Σr and p2 (Σr ) are equidimensional. From dim S (S u,v ) = 5 and our assumption that N ≥ 5, we are able to choose a  (D − N)-plane which intersect S (S u,v ) in any one and exactly one point. Therefore, we can find P so that P⊥ intersects S (S u,v ) in one point outside T (S u,v ) ∪ Z2 . This provides an example of (2.27) for m = r = 1. It follows that p2 has degree one, and the image is rational since σ l1 , l1 0 is rational by Corollary 2.22.   2.3.5 Proof of Theorem 2.17 Lemma 2.25. Assume D ≥ N ≥ 5, and assume the existence of a degree D rational scroll S ⊂ PN with isolated singularities which has at least r singularities. Then Hu,v r has codimension at most r(N − 4) in Hu,v . For r = 1, Hu,v 1 is unirational of codimension exactly N − 4. 85 Proof. We have the following diagram π / –(N, D + 1) Hu,v p p2  Σr / G(N, D + 1).   r By definition, Hu,v = π p−1 (p2 (Σr )) . 1 By Lemma 2.24, p2 (Σ1 ) is rational, which implies that Hu,v is unirational. It’s clear that p2 (Σr ) and p−1 (p2 (Σr )) have the same codimension. On the other hand,    p−1 (p2 (Σr )) and π−1 π p−1 (p2 (Σr )) have the same dimension since both contain an open dense subset consisting of the projections which generate r singularities, so the codimen- sion of p−1 (p2 (Σr )) is the same as its image through π. Therefore, p2 (Σr ) and Hu,v r have the same codimension in their own ambient spaces, and the results follows from Lemma 2.24.  Lemma 2.25 is the special case of Theorem 2.17 when restricting to the locus of a particular type on the Hilbert scheme. The next lemma shows that a general S ∈ HDr deforms equisingularly between different types under the assumption rN ≤ (D + 2)2 − 1. Hence the dimension estimate made by Lemma 2.25 can be extended regardless of the types. Lemma 2.26. Assume (2.27) and rN ≤ (D + 2)2 − 1, then Hu,v r = Hu,v ∩ Hu+k,v−k r for 0 ≤ 2k < v − u. r r r r Proof. It is trivial that Hu,v ⊃ Hu,v ∩ Hu+k,v−k . To prove that Hu,v ⊂ Hu,v ∩ Hu+k,v−k , it is r sufficient to show that a generic element in Hu,v deforms equisingularly to an element in Hu+k,v−k .     If (u, v) = b D2 c, d D2 e then there is nothing to prove, so we assume (u, v) , b D2 c, d D2 e . r r The elements satisfying (2.27) form an open dense subset of Hu,v . Let S ∈ Hu,v be one of 86 them, and assume S is the image of F  S u,v ⊂ PD+1 projected from some (D − N)-plane Q. By hypothesis, F has r secants γ1 , ..., γr incident to Q. Assume γ j ∩ F = {x j , y j } for j = 1, ..., r. H 1 (T PD+1 |F ) = 0 by Lemma 2.4, so the short exact sequence 0 → T F → T PD+1 |F → NF/PD+1 → 0 induces the exact sequence   0 → H 0 (T F ) → H 0 (T PD+1 |F ) → H 0 NF/PD+1 → H 1 (T F ) → 0.   By Lemma 2.2, h1 (F, T F ) = h1 P1 , OP1 (u − v) = v − u − 1, the same as the codimension of Hu,v in HD , thus a deformation normal to Hu,v is induced from an element in H 1 (T F ). In order to prove that the deformation is equisingular, it is sufficient to prove that for all   F ∈ H 1 (T F ) and its lift S ∈ H 0 NF/PD+1 , there exists α ∈ H 0 (T PD+1 |F ) such that the vectors S(x j ) + α(x j ) ∈ T PD+1 ,x j and S(y j ) + α(y j ) ∈ T PD+1 ,y j keep γ j contact with Q for j = 1, ..., r, so that S + α is a lift of F representing an embedded deformation which preserves the incidence of the r secants to Q. Note that for arbitrary p ∈ PD+1 , the tangent space T PD+1 ,p  Hom p, p⊥  p⊥ can be  considered as a subspace of PD+1 . We identify a point in PD+1 with its underlying vector. Let γ = γ j for some j, and let {x, y} = γ ∩ S u,v with x = (x1 , ..., xD+1 ) and y = (y1 , ..., yD+1 ). The condition that S(x) + α(x) and S(y) + α(y) keep γ contact with Q is equivalent to the condition that the set of vectors consisting of x + S(x) + α(x), y + S(y) + α(y) and the basis of Q is not independent. One can compute that h0 (T PD+1 |F ) = (D+2)2 −1 by the Euler exact sequence 0 → OF → OF (1)⊕(D+2) → T PD+1 |F → 0 and Lemma 2.1. Suppose H 0 (T PD+1 |F ) has basis e1 , ..., e(D+2)2 −1 , we write the evaluation of ei at p as ei (p) = (ei (p)1 , ..., ei (p)D+1 ). Let α = i≥1 αi ei , S(x) = P   = = qi, j as a (D − N + 1) × (D + 2)-matrix. P P c e i≥1 i i (x) and S(y) d e i≥1 i i (y), also write Q 87 Then the dependence condition is equivalent to the condition that the (D − N + 3) × (D + 2)- matrix    α0 x + α0 S(x) + α(x)      Aγ =  α0 y + α0 S(y) + α(y)      Q     α x + P (α c + α ) e (x) ... α0 xD+1 + (α0 ci + αi ) ei (x)D+1  P  0 0 0 i i i 0    α y + (α d + α ) e (y) P ... α0 yD+1 + (α0 di + αi ) ei (y)D+1  P   0 0 0 i i i 0  =   ...  q0,0 q0,D+1  .. ..   . .       ...   qD−N,0 qD−N,D+1 has rank at most D − N + 2. Here we homogenize the first two rows by α0 , so that the matrix defines a morphism   2 Aγ : P C ⊕ H 0 (T PD+1 |F )  P(D+2) −1 → P(D−N+3)(D+2)−1 (α0 , ..., α(D+2)2 −1 ) 7→ Aγ . Let MD−N+2 ⊂ P(D−N+3)(D+2)−1 be the determinantal variety of matrices of rank at most   D − N + 2. Then Aγ −1 (MD−N+2 ) ⊂ P C ⊕ H 0 (T PD+1 |F ) is an irreducible and nonde- generate subvariety of codimension N, whose locus outside α0 = 0 parametrizes those α ∈ H 0 (T PD+1 |F ) such that S + α preserves the incidence between γ and Q. It follows that the intersection rj=1 Aγ j −1 (MD−N+2 ) is nonempty by the hypothesis T rN ≤ (D + 2)2 − 1. Moreover, it is not contained in the hyperplane α0 = 0 for a generic r S ∈ Hu,v . Indeed, if this doesn’t hold, then the limit case γ1 = ... = γr should also be inside the hyperplane α0 = 0. However, the intersection in that case is a multiple of a nondegenerate variety, a contradiction. As a result, for a generic S ∈ Hu,v r we can find α from rj=1 Aγ j −1 (MD−N+2 ) which lies on {α0 = 1} = H 0 (T PD+1 |F ), so that S + α preserves T 88 the incidence condition between γ1 , ..., γr and Q.  Now we are ready to finish the proof of Theorem 2.17. Note that HDr = u+v=D Hu,vr S . By Lemma 2.26 [ [   r Hu,v = Hu,v ∩ Hb D c,d D e = HD ∩ HbrD c,d D e = HbrD c,d D e . r 2 2 2 2 2 2 u+v=D u+v=D Therefore HDr = HbrD c,d D e , and the result follows from Lemma 2.25 with (u, v) = (b D2 c, d D2 e). 2 2 Chapter 3 Rationality and unirationality of cubic fourfolds For a smooth projective variety X of dimension n, a unirational parametrization for X is a generically finite map ρ : Pn d X. We say that X is rational if it admits a unirational parametrization of degree one. Other- wise, we say that X is irrational. Kuznetsov has conjectured that a cubic fourfold is rational if and only if it has an associated K3 surface [Kuz10, AT14]. These cubics form a union of countably many irreducible divisors Cd in the moduli space, indexed by an infinite sequence of positive integers starting with d = 14, 26, 38, 42, .... It is known that a general X ∈ Cd is rational for d = 14, 26, 38. The rationality in the case d = 14 is established in the work done by Morin [Mor40] and subsequently by Fano [Fan43] in the 1940s. The cases d = 26 and 38 remained open until 2017 by Russo and 89 90 Staglian`o [RS17]. It is well-known that a cubic fourfold admits a unirational parametrization of degree two (Lemma 3.13). How common are unirational parametrizations of odd degree? Ob- viously, a rational cubic fourfold admits such a parametrization. In addition to the cases d = 14, 26, 38, there are a few examples known to satisfy the property also. For instance, we will show that a general X ∈ Cd for d = 42 admits a unirational parametrization of degree 13. In Section 3.1, we review some basic facts on cubic fourfolds and their relations to K3 surfaces. We show that every X ∈ C26 is rational in Section 3.2. Section 3.3 focuses on the unirational parametrization of odd degree, especially in the case d = 42. 3.1 Cubic fourfolds with associated K3 surfaces Cubic fourfolds are closely related to K3 surfaces in many aspects. In this section, we start with the example in Morin and Fano’s works. Then we introduce a general framework which includes the example as a special case. In certain special cases, we exhibit how the associated K3 surfaces induce rational scrolls in the cubic fourfolds, which is essential in conquering the rationality and unirationality problems in the known examples. 3.1.1 Pfaffian cubic fourfolds We formulate the example in a relatively modern form provided by Beauville and Donagi [BD85]. Let V  C6 be a vector space. We identify the space of two forms 2 ^ V ⊂ End(V ∨ , V) 91 as the subspace of antisymmetric linear maps. Thus every M ∈ 2 V is of even rank, V √ and its Pfaffian Pf(M) = det(M) is well-defined as a cubic in its entries. Recall that rk(M) ≤ 2 if and only if the associated two-form splits. Thus we have the stratification by ranks 2 ^ Gr(2, V) ⊂ Pfaff(V) ⊂ P( V) where    ^2  Pfaff(V) =  =   [M] ∈ V) : Pf(M) 0    P(     is a cubic hypersurface in P( 2 V)  P14 . We also have the dual strata with respect to the V dual space V ∨ : 2 ^ Gr(2, V ∨ ) ⊂ Pfaff(V ∨ ) ⊂ P( V ∨ ). Let L ⊂ P( 2 V) be a general dimension five subspace, which corresponds to a dimen- V sion eight subspace L⊥ ⊂ P( 2 V ∨ ). Then V X = L ∩ Pfaff(V) ⊂ L  P5 is a cubic fourfold. On the other hand, S = L⊥ ∩ Gr(2, V ∨ ) ⊂ L⊥  P8 is a K3 surface of degree 14 [Muk88]. Beauville and Donagi [BD85] established the following properties: 1. For each hypersurface H ⊂ P(V ∨ ), the map X d H : [M] 7→ [ker(M)] ∩ H 92 is birational. 2. The variety F1 (X) of lines on X and the Hilbert scheme S [2] of length two sub- schemes on S are isomorphic. 3. Let [P] ∈ S ⊂ Gr(2, V ∨ ) be any point, represented by a linear subspace P ⊂ V ∨ . Then the locus RP = {[M] ∈ X : ker(M) ∩ P , 0} is a quartic scroll in P5 , realized as the image of P1 × P1 under the linear series |OP1 ×P1 (1, 2)|. The scroll RP and the square h2 of hyperplane class generate a lattice K = Zh2 ⊕ ZRP ⊂ H 4 (X, Z). It is straitforward to verify that the self-intersection Z RP · RP = c(RP )−1 c(X) = 10, RP where c(RP ) and c(X) are the total Chern classes of RP and X, respectively. Hence the lattice K has the intersection table h2 R P h2 3 4 RP 4 10, whence K has discriminant 3 · 10 − 42 = 14. The existence of the K3 surface S and its relations to X can be generalized to certain 93 special cubic fourfolds, which we introduce in the following. 3.1.2 Special cubic fourfolds Smooth cubics in |OP5 (3)| are stable objects under the action of PGL(6) in the sense of geometric invariant theory [MFK94, Prop. 4.2]. They form an open subset U ⊂ |OP5 (3)| complement to an irreducible divisor [EH16, Prop. 7.1]. The moduli space of cubic four- folds is the geometric quotient C = [U/PGL6 ] , which is a Deligne-Mumford stack with quasi-projective coarse moduli space [MFK94, §4.2]. In particular, we have dim(C) = dim(U) − dim(PGL6 ) = 55 − 35 = 20. The Torelli theorem [Voi86] implies that, for a general cubic fourfold X ∈ C, the group of its integral Hodge classes of degree 4 Hdg4 (X, Z) = H 4 (X, Z) ∩ H 2 (Ω2X ) is generated by the square h2 of hyperplane class. We say that X is special if there exists a rank two saturated sublattice h2 ∈ K ⊂ Hdg4 (X, Z), called a labelling of X. Because the integral Hodge conjecture is valid for cubic fourfolds [Voi13, Th. 1.4], every labelling is spanned by h2 and some algebraic surface R ⊂ X. Hence X ∈ C is special if and only if it contains an algebraic surface not homologous to a complete intersection. The discriminant of a labelling K = Zh2 ⊕ ZR is defined as the 94 determinant of the intersection matrix    h2 · h2 h2 · R   .     R · h2 R · R  3.1.3 Associated K3 surfaces Let X be a special cubic fourfold admitting a labelling K. Then the Hodge numbers of the orthogonal complement K ⊥ ⊂ H 4 (X, Z) take the form 0 1 19 1 0 of a K3 surface. We say that a polarized K3 surface (S , f ) is associated to (X, K) if there is a Hodge isometry H 2 (S , Z)(−1) ⊃ f ⊥ ∼ / K ⊥ ⊂ H 4 (X, Z). Theorem 3.1 ([Has00]). Special cubic fourfolds admitting labellings of discriminant d form an irreducible divisor Cd ⊂ C, which is non-empty if and only if (3.1) d≥8 and d ≡ 0, 2 (mod 6). Moreover, X ∈ Cd admits an associated K3 surface S , polarized of degree d, if and only if (3.2) d is not divisible by 4, 9, or any odd prime p ≡ 2 (mod 3). 95 Assume further that (3.3) d = 2(n2 + n + 1), n≥2 and that X is generically chosen. Then the variety F1 (X) of lines on X is isomorphic to the Hilbert scheme S [2] of length two subschemes on S . For example, the Pfaffian cubic fourfolds correspond to the case d = 14 = 2(22 +2+1). The other discriminants admitting associated K3 surfaces and the relation F1 (X)  S [2] include d = 26 and 42, which correspond to n = 3 and 4 in (3.3) respectively. Remark 3.2. A general X ∈ Cd admits two associated K3 surfaces when d ≡ 2 (mod 6) (e.g. d = 42) and one otherwise [Has16, Cor. 25]. 3.1.4 Cycles on the Hilbert square We review some basic facts on the Hilbert square of a K3 surface and refer to [HT10] for the details. See also [FV18, §2]. Let S be a polarized K3 surface with Pic(S ) = Z f and f 2 = d, and let S [2] be the Hilbert scheme of length two subschemes on S . For every point p ∈ S , define n o ∆ p = ξ ∈ S [2] : supp(ξ) = p . The elements of ∆ p are determined by the directions tangent to p, so ∆ p is canonically isomorphic to the tangent space P(T S p ). In particular, it is a rational curve on S [2] . The union of all ∆ p form a divisor [ ∆= ∆ p  P(T S ) p∈S 96 whose cohomology class is divisible by two. Define [∆] δ= ∈ H 2 (S [2] , Z) and δ p = [∆ p ] ∈ H2 (S [2] , Z). 2 Let C ∈ | f | be a general curve and assume that p ∈ S is general. Consider the divisor n o fC = ξ ∈ S [2] : supp(ξ) ∩ C , ∅ , and the curve n o F p = ξ = p + x ∈ S [2] : x ∈ C , and define f = [ fC ] ∈ H 2 (S [2] , Z) and f p = [F p ] ∈ H2 (S [2] , Z). It is straitforward to establish the intersection pairing from the construction: f δ fp d 0 δ p 0 −1. On the other hand, H 2 (S [2] , Z) is endowed with a nondegenerate integral quadratic form q, known as the Beauville-Bogomolov form [Bea83]. The lattice (H 2 (S [2] , Z), q) has the orthogonal decomposition H 2 (S [2] , Z)  H 2 (S , Z) ⊕⊥ Zδ, q(δ, δ) = −2, where the pairing on H 2 (S , Z) coincides with the intersection product. 97 3.1.5 Scrolls from the Hilbert square Assume that X ∈ Cd satisfies condition (3.3) and let S be an associated K3 surface S . In this case, the locus n o ∆ p = ξ ∈ S [2] : supp(ξ) = p corresponds to a rational curve on F1 (X) via the isomorphism F1 (X)  S [2] , and thus swipes out a rational scroll R p ⊂ X. By varying p, we obtain a two-dimensional family of scrolls in X parametrized by S . Proposition 3.3. The class h2 and the scroll R p generate a rank-two saturated sublattice in H 4 (X, Z) with intersection pairing h2 Rp h2 3 2n + 1 R p 2n + 1 2n2 + 2n + 1, where n is the integer satisfying d = 2(n2 + n + 1). Furthermore, if R p is singular along a zero-dimensional locus, then it has n(n − 2) nodes. Proof. The incidence correspondence between X and F1 (X) induces the Abel-Jacohi map [BD85] α : H 4 (X, Z) ∼ / H 2 (F1 (X), Z)(−1) compatible with the Hodge structures. The class α(h2 ) = g coincides with the polarization on F1 (X) under the pl¨ucker embedding F1 (X) ⊂ Gr(2, 6) ,→ P14 . Consider g as a class in H 2 (S [2] , Z) under the isomorphism F1 (X)  S [2] . Then we have 98 g = 2 f − (2n + 1)δ [Has00, §6], and thus h2 · R p = g · δ p = 2( f · δ p ) − (2n + 1)(δ · δ p ) = 2n + 1. The self-intersection of R p is computed from d = 3(R p · R p ) − 2(h2 · R p ). The normalization of R p is a rational normal scroll Σ ⊂ P2n+2 , projectively equivalent to the embedding of the Hirzebruch surface |hΣ := g + uγ| : Fm ,→ P2n+2 . Here g denotes the class of a section and γ denotes the fiber class. Note that 2n + 1 = deg(R p ) = deg(Σ) = h2Σ = m + 2u. Let π : Σ → R p be the normalization. By the double point formula [Ful98, Th. 9.3], Z ! 1 (3.4) | nodes on R p | = Rp · Rp − −1 c(Σ) · π c(X) . ∗ 2 Σ We have π∗ c(X) = π∗ (c(P5 ) · c(NX/P5 )) = (1Σ + hΣ )6 (1Σ + 3hΣ )−1 = (1Σ + 6hΣ + 15h2Σ )(1Σ − 3hΣ + 9h2Σ ) = 1Σ + 3hΣ + 6h2Σ , and c(Σ)−1 = (1Σ − KΣ + χ(Σ))−1 = 1Σ + KΣ + (KΣ2 − χ(Σ)). 99 Thus c(Σ)−1 · π∗ c(X) = 3hΣ KΣ + (KΣ2 − χ(Σ)) + 6h2Σ R Σ = 3(g + uγ)(−2g + (m − 2)γ) + (−2g + (m − 2)γ)2 − χ(Σ) + 6h2Σ = 3(−m − 2u − 2) + (8) − 4 + 6h2Σ = 3h2Σ − 2 = 6n + 1. Therefore, the number of nodes on R p equals 1 1 ((2n2 + 2n + 1) − (6n + 1)) = (2n2 − 4n) = n(n − 2) 2 2 provided that R p has isolated singularities. Note that this result is independent of the type of Σ.  3.2 Rationality of cubic fourfolds of discriminant 26 The rationality of a general cubic fourfold of discriminant 26 or 38 has been proved by Russo and Staglian`o [RS17], where they realize the cubic as a rational section of some conic bundle over a rational fourfold. Here we exhibit a proof in the case of discriminant 26, following basically the approach in the first version of [RS17] with slightly different details. 3.2.1 Septic scrolls with three nodes A general X ∈ C26 admits an associated K3 surface S , and the variety of lines F1 (X) is isomorphic to the Hilbert square S [2] . The locus n o ∆ = ξ ∈ S [2] : supp(ξ) = p ∈ S  P(T S ) 100 induces a family of scrolls R p ⊂ X parametrized by p ∈ S . Prop. 3.3 implies that R p has degree 7, and is singular along three nodes provided that it has isolated singularities. Fix a balanced rational normal scroll Σ ⊂ P8 and let Sec(Σ) be its secant variety. Let H denote the Hilbert scheme of 3-nodal septic scrolls in P5 . Then a general R ∈ H is projectively equivalent to the projection of Σ from three points a1 , a2 , a3 ∈ Sec(Σ). Set R = H/PGL(6). Farkas and Verra [FV18] prove that R ∼ / Sym3 (Sec(Σ))/Aut(Σ), and there is an incidence correspondence X = {(R, X) : R ⊂ X} (3.5) π1 π2 v ( R C26 where π2 : X → C26 is fibered in the associated K3 surfaces. They also show that the defining ideal IR for a general R ∈ R in P5 is generated by 13 cubics [FV18, Prop. 3.4]. Thus π1 : X → R is birational to a projective bundle generically fibered in |IR (3)|  P12 . Let (z0 , ..., z8 ) be homogeneous coordinates on P8 . Then, up to the PGL(9)-action, Σ ⊂ P8 is defined by the 2 × 2-minors of the matrix    z z z z z z z   0 1 2 3 5 6 7  (3.6)  .  z z z z z z z  1 2 3 4 6 7 8 101 Moreover, Sec(Σ) ⊂ P8 is defined by the 3 × 3-minors of the matrix    z0 z1 z2 z5 z6      (3.7)  z z z z z  .  1 2 3 6 7     z z z z z  2 3 4 7 8 It follows that Sec(Σ) has dimension 5 [CJ96, Lemma 3.1], and its singular locus is exactly Σ [Eis88, Cor. 3.3]. Example 3.4. Here we provide the data of an explicit example which is used in proving Prop. 3.5 and Lemma 3.7. Our computations are done with Singular [DGPS15]. First we define the scroll Σ ⊂ P8 via (3.6). Choose a1 = (1, 0, 1, 0, 1, 0, 0, 0, 0) a2 = (1, 0, 0, 0, 0, 1, 0, 0, 1) a3 = (0, 1, 0, 0, 0, 0, 2, 0, 0) from Sec(Σ) and complete them to a nonsingular matrix     1 0 1 0 1 0 0 0 0       1 0 0 0 0 1 0 0 1       0 1 0 0 0 0 2 0 0       1 2 8 9 4 3 6 7 5   T =  3 1 2 8 5 4 9 7 6  .        2 3 5 1 7 8 6 9 4       1 7 2 9 3 5 6 4 8       6 5 9 7 1 2 3 8 4      4 2 3 9 8 6 1 7 5  102 Then z = T z0 defines a coordinate transformation of P8 , where z = (z0 , ..., z8 ) and z = (z00 , ..., z08 ) are considered as column vectors. It maps Σ to Σ0 and a1 , a2 , a3 to a01 = (1, 0, 0, 0, 0, 0, 0, 0, 0) a02 = (0, 1, 0, 0, 0, 0, 0, 0, 0) a03 = (0, 0, 1, 0, 0, 0, 0, 0, 0). Under the new coordinates the projection πΛ : P8 d P5 from Λ = span(a1 , a2 , a3 ) ⊂ P8 is defined by (x0 , ..., x5 ) = (z03 , ..., z08 ). Then we set R = πΛ (Σ0 ) ⊂ P5 . Below we provide the data of a point x ∈ P5 and a hyperplane x ∈ H x ⊂ P5 which are also needed in the proofs: x = (6, 4, 0, 5, 8, 2), H x = {x2 = 0}. 3.2.2 The birational transformation Let R ∈ R be a general 3-nodal septic scroll in P5 and let IR be the defining ideal. Then the linear system |IR (3)| defines a rational map F : P5 / P12 whose base scheme coincides with R. We denote by W its image in P12 . Proposition 3.5. The map F is a birational transformation, i.e. it is birational onto its 103 image W ⊂ P12 . Proof. The defining ideal for the scroll R of Example 3.4 has linear syzygies. Hence the corresponding map F has linear fibers over a general closed point by [Ver01, Prop. 2.8 & Remark 2.9]. The ideal of the fiber F −1 (F(x))\R over x can be computed by the quotient   n o IF −1 (F(x)) : IR = r ∈ C[x] : rIR ⊂ IF −1 (F(x)) where C[x] denotes the basering. We verify that it is a simple point, so F is a birational transformation. These properties are preserved by deformation so a general such construc- tion also gives a birational transformation.  We want to know how a curve C ⊂ P5 is transformed onto its image via F. Let Z = BlR P5 . We have Pic(Z) = ZH ⊕ ZE, where H comes from the hyperplane class on P5 and E is the exceptional divisor. Then the linear series associated to the class M = 3H − E. define a morphism F˜ : Z → W resolving F. Thus we obtain a commutative diagram Z F˜  " P5 F / W ⊂ P12 . Let C ⊂ P5 be a curve of degree r and n-secant to R. This is equivalent to saying that its strict transform C˜ ⊂ Z has the intersection numbers C˜ · H = r and C˜ · E = n. 104 It follows that the image F(C) ⊂ W ⊂ P12 is a curve of degree C˜ · M = C˜ · (3H − E) = 3r − n. In particular, a line bisecant to R or a conic 5-secant to R is mapped to a line on W. Lemma 3.6. Through a general point x ∈ P5 there pass exactly seven lines bisecant to R. Proof. The statement holds if and only if the projection from x: π x : P5 d P4 produces seven additional nodes on the image T = π x (R). Since R already has three nodes, this is equivalent to that the projection from a general 3-plane P ⊂ P8 πP : P8 d P4 maps Σ ⊂ P8 to a scroll T ⊂ P4 singular along ten nodes. The latter condition can be verified straightforwardly by the double point formula [Ful98, Th. 9.3] Z ! 1 | nodes on T | = [T ]P4 2 − −1 c(Σ) · π∗P c(P ) . 4 2 Σ  Lemma 3.7. Through a general point y ∈ W ⊂ P12 there pass eight lines contained in W. Seven of them correspond to lines bisecant to R via F, while the remaining one corresponds to a conic 5-secant to R. Proof. The point y ∈ W determines a 5-plane T y,W  P5 ⊂ P12 tangent to W at y. It can be 105 computed explicitly from the Jacobian of F at x = F −1 (y): ∂F ∂F ! T y,W = span (x), . . . , (x) ∂x0 ∂x5 where (x0 , ..., x5 ) are homogeneous coordinates on P5 . Let V denote the singular locus of F −1 (T y,W ) ⊂ P5 . Note that a line l ⊂ W passing through y belongs to the singular locus of W ∩ T y,W . This implies that F −1 (l) ⊂ V. We verify on Example 3.4 that 1. V is a curve of degree 9. 2. V intersects a general X ∈ |IR (3)| in eight points away from R. 3. There’s a hyperplane H x ⊂ P5 passing through x which intersects V in a single point away from x. Let V˜ ⊂ Z denote the strict transform of V. Then (1) and (2) imply that V˜ is a curve such that V˜ · H = 9 and V˜ · M = 8. By Lemma 3.6, there are seven lines l1 , ..., l7 ⊂ P5 passing through x and bisecant to R. They are mapped to lines on W passing through y and thus contained in V. It follows that V can be decomposed into curves V = C ∪ l1 ∪ · · · ∪ l7 such that the strict transform C˜ ⊂ Z of C satisfies C˜ · H = 2 and C˜ · E = C˜ · (3H − M) = 6 − 1 = 5. Hence C is a conic 5-secant to R, which passes through x by (3). This completes the 106 proof.  3.2.3 The rational parametrization Lemma 3.7 implies that the Fano variety of lines on W is a fourfold and is a union of two components F1 (W)  L ∪ H, where L corresponds to the bisecant lines and H corresponds to the 5-secant conics. For every cubic X ∈ |IR (3)|, we can define a rational map κX : H / X which maps l ⊂ W to the point F −1 (l) ∩ X 0 \R. Lemma 3.7 implies that κX is birational. In particular, we’ve proved the following Proposition 3.8. A general X ∈ C26 contains a 3-nodal septic scroll R such that all cubics cutting R in P5 are birational to each other. Note that every member of |IR (3)| is irreducible since R is not contained in a hyperplane or a quadric. Remark 3.9. The component H provides an explicit example of a minimal rational com- ponent in the sense of [HM04]. In fact, the birationality can be proved without knowing precisely the degree and the number of intersection points with R of the curves parametrized by H [RS17, Th. 1]. It follows that every member of |IR (3)| is rational if one of them is rational. We find rational cubics from the singular ones. Recall that singular cubics are determined by the discriminant locus D ⊂ |OP5 (3)| [EH16, §7.1]. It is an irreducible hypersurface in 107 |OP5 (3)|  P55 [EH16, Prop. 7.1] of degree 192 in our case [GKZ08, LZ13]. Since |IR (3)| contains a smooth cubic, singular cubics form a hypersurface in |IR (3)|. Lemma 3.10. There exists a rational singular cubic Y ∈ |IR (3)|. Proof. By Lemma 3.6, we can find a point x ∈ P5 which admits a line l 3 x bisecant to R. We can also find Y ∈ |IR (3)| singular at x. Indeed, such cubics are defined by the vanishing of the Jacobians at x and thus of codimension at most six in |IR (3)|  P12 . When x is a double point, a general line passing through x intersects Y in precisely one other point. So ∼ the projection from x defines a birational map Y d P4 . Assume that x is a triple point. Then every line joining x and one other point on Y is contained in Y. Thus Y is a cone with vertex x over an irreducible cubic threefold Y 0 ⊂ P4 . The projection from x maps R ⊂ Y onto a rational surface R0 ⊂ Y 0 and l to a node x0 ∈ R0 . The tangent cone at x0 to R0 spans the ambient P4 , so Y 0 is singular at x0 . If x0 is a double point then Y is rational. If x0 is a triple point, then Y 0 is a cone over an irreducible surface Y 00 ⊂ P3 via a similar argument. The projection from x0 restricts to a rational map π0 : R0 d Y 00 , which is generically finite since the lines in R0 passing through x0 consists of two rulings [Lai17, §1.5]. So π0 is dominant, which implies that Y 00 is unirational, and thus is rational by Castelnuovo’s theorem. Therefore, Y 0 is rational and thus Y is rational.  As a result, the component H is rational and thus every X ∈ |IR (3)| is rational. Hence a general X ∈ C26 is rational.. Theorem 3.11. Every X ∈ C26 is rational. Proof. We work on the space |OP5 (3)| since it has a universal family n o E = (p, X) ∈ P5 × |OP5 (3)| : p ∈ X 108 Let V ⊂ |OP5 (3)| denote the preimages of C26 and let V ◦ ⊂ V be the open subset parametrizes rational cubics. Then it’s equivalent to show that the fibers of E over V\V ◦ are all rational. Let x ∈ V\V ◦ be any closed point. By taking hyperplane sections in |OP5 (3)|  P55 followed by a resolution of singularities, we can get a morphism ι : B → V for some smooth connected curve B, such that x lies in the image and that ι−1 (V ◦ ) is dense in B. Assume that x = ι(x0 ) for some closed point x0 ∈ B. The pullback ι∗ E is birational to H × B over B and thus rational over the function field K(B). This allows us to apply [KT17, Th. 1] to conclude that (ι∗ E) x0 is rational, and thus E x  (ι∗ E) x0 is rational.  Remark 3.12. It is known that the set of rational cubic fourfolds is a countable union of locally closed subsets of the moduli space [Has16, §2.4]. Prop. 3.11 implies that it is in fact a countable union of closed subsets. De Fernex and Fusi [dFF13] have proved a similar result for fibers of relative dimension three. 3.3 Unirational parametrizations of odd degree The unirationality of a smooth projective variety X is closely related to the property of CH0 (X), the Chow group of 0-cycles on X. For a positive integer N, we say that CH0 (X) is universally N-torsion if the degree map deg : CH0 (XF ) → Z is surjective and has kernel killed by N for all base extension XF = X ×C F. If X admits a unirational parametrization of degree %, then CH0 (X) is universally %-torsion [CTC79, Prop. 6.4]. Lemma 3.13. Let X ⊂ Pnk be a cubic hypersurface over an arbitrary field k. If X contains a k-rational line L, then X admits a unirational parametrization of degree two. 109 Proof. Consider the incidence correspondence Y = (l, p) : l ⊂ Pn is a line tangent to X at p ∈ L ,  which is canonically isomorphic to the restriction of the projectivized tangent bundle P(T X) to L  P1 . In particular, Y is rational of dimension (n − 1). A general line l tangent to X at p intersects X in exactly one other point. This defines a rational map ρ : Y d X : (l, p) 7→ l ∩ X\{p}. We prove that ρ has degree two. Let q ∈ X be a general point. Then span(L, q) ∩ X = L ∪ C where C is a conic and we have L ∩ C = {p1 , p2 }. Let l1 = span(p1 , q) and l2 = span(p2 , q). Then ρ−1 (q) = {(l1 , p1 ), (l2 , p2 )}.  It follows that CH0 (X) is universally 2-torsion for a cubic fourfold X. Furthermore, for a special cubic fourfold X ∈ Cd with 4 - d, Voisin [Voi17] has proved that CH0 (X) is universally 1-torsion. Hence it is natural to ask if X admits odd-degree unirational parametrizations. In fact, Nuer’s examples [Nue16] together with Prop. 3.17 provide such parametrizations for d = 14, 18, 26, 30, 38. Our goal is to improve this list to d = 42. Theorem 3.14. C42 is uniruled, and a general X ∈ C42 admits a unirational parametriza- tion of degree 13. 110 3.3.1 Degree-9 scrolls with eight nodes Let H9 be the component in the Hilbert scheme corresponding to degree-9 rational scrolls. The closure of the locus parametrizing 8-nodal scrolls form a subscheme H98 ⊂ H9 . Prop. 2.11 and Th. 2.17 in Ch. 2 implies that dim(H98 ) ≤ 8. Note that H98 parametrizes nonre- duced schemes by definition. In the following, we mark an element S ∈ H98 with an overline and denote by S = red(S ) its underlying reduced subscheme. Let U ⊂ |OP5 (3)| be the open subset parametrizing smooth cubics. Define n o Z = (S , X) ∈ H98 × U : S ⊂ X . By Prop. 2.12, Ch. 2, Z is nonempty and it contains a pair (S , X) where S has isolated singularities and X is smooth. Let U42 ⊂ U be the locus parametrizing cubic fourfolds of discriminant 42, that is, the preimage of C42 under the quotient C = [U/PGL6 ]. Lemma 3.15. The right projection Z → U factors through U42 . Proof. Let (S , X) ∈ Z be an element. The double point formula (3.4) and the same calculation following it imply that 1 8 = ([S ]2X − 27) ⇒ S · S = 41. 2 Let h be the hyperplane class on X. Then the intersections between h2 and S in X are h2 S h2 3 9 S 9 41. So X has discriminant 3 · 41 − 92 = 42.  111 3.3.2 Unirational parametrizations of degree 13 Theorem 3.16. In the diagram Z p1 p2 ~ H98 U42 / C42 , 1. Z dominates U42 . Therefore a general X ∈ C42 contains a degree-9 rational scroll with 8 nodes and smooth otherwise. 2. C42 is uniruled. Proof. Let (S , X) ∈ Z be the example in Prop. 2.12, Ch. 2. From the exact sequence 0 → IS (3) → OP5 (3) → OS (3) → 0 and the fact that h0 (P5 , IS (3)) = 6, we get 6 − h1 (P5 , IS (3)) = h0 (P5 , OP5 (3)) − h0 (S , OS (3)). Let Σ ⊂ P10 be the rational normal scroll which normalizes S . Then the sections in H 0 (S , OS (3)) coincide with the sections in H 0 (Σ, OΣ (3)) which cannot distinguish the preimage of a node. We have h0 (F, OF (3)) = 58 by Lemma 2.1, Ch. 2, thus h0 (S , OS (3)) = 58 − 8 = 50. Hence the above equation becomes 6 − h1 (P5 , IS (3)) = 56 − 50 = 6. Therefore h1 (P5 , IS (3)) = 0. 112 It follows that p1 : Z → H98 is birational to a projective bundle fibered in |IS (3)|  P5 . We have dim H9 = 59 by Prop. 2.16, Ch. 2, and thus dim H98 ≥ 59 − 8 = 51 by Th. 2.17, Ch. 2. Hence dim(Z) ≥ 51 + 5 = 56. On the other hand, Prop. 2.12 (3), Ch. 2, implies that Z → p(Z) ⊂ U42 has generic fiber of dimension ≤ 2. It follows that dim(p2 (Z)) ≥ 56 − 2 = 54. Since U42 ⊂ U is a divisor and thus has dimension 54 also, Z must dominate U42 . Now we show that C42 is uniruled. For a general S ∈ H98 , there exists a smooth cubic X ∈ |IS (3)|. On the other hand, the same argument as in the proof of Lemma 3.10 implies that there exists a singular cubic Y ∈ |IS (3)|. X and Y span a line L ⊂ |OP5 (3)| such that L◦ = L ∩ U42 3 X. Because X and Y cannot be projectively equivalent, L◦ is not contracted via the map U42 → C42 . For a general (S , X) ∈ Z, we can find such a line passing through X ∈ C42 . Hence C42 is uniruled.  Proposition 3.17. [Has16, Prop. 38] Let X be a cubic fourfold and S ⊂ X be a rational surface. Suppose S has isolated singularities and smooth normalization, with invariants D = deg S and section genus gH . Assume that D(D − 2) [S ]2X % := + (2 − 2gH ) − > 0. 2 2 Then X admits a unirational parametrization ρ : P4 d X of degree %. Idea of the proof. Because S is rational, its symmetric square Sym2 (S ) is also rational. We have a unirational parametrization ρ : Sym2 (S ) d X : s1 + s2 7→ x defined via span(s1 , s2 ) ∩ X = {s1 , s2 , x}. 113 The degree of ρ equals the number of lines through a general point of X which is secant to R. Consider the projection π x : P5 d P4 from a general point x ∈ X and let Sˆ := π x (S ) ⊂ P4 . Then the number is the same as the number of nodes on Sˆ produced via the projection, which equals the difference between the numbers of nodes on Sˆ and S , and thus can be computed by the double point formula (3.4). The hypothesis on the singularity of S is used to confirm that π x is unramified on S and that the double point formula works.  Corollary 3.18. A general X ∈ C42 has an unirational parametrization of degree 13. Proof. 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