Moduli spaces play a central role in algebraic geometry. In this thesis we study the geometry of two particular moduli spaces. In Part I we study the Hilbert scheme of points of an algebraic variety which is a partial compactification of the configuration space of distinct unordered points. The motivic Hilbert zeta function is the generating series for classes in the Grothendieck ring of varieties of the Hilbert schemes of points. It is a rich invariant which in the case of a smooth curve is a motivic enhancement of the Hasse-Weil zeta function. We show that the motivic Hilbert zeta function of a singular curve is rational. Furthermore, we show that the Euler characteristic specialization of the Hilbert zeta function is constructible in flat families of curve singularities. In Part II we study the stable pair compactifications of the moduli space of elliptic fibrations furnished by the minimal model program. Motivated by Hassett's weighted pointed stable curves, we construct compact moduli spaces parameterizing weighted stable elliptic surfaces -- elliptic fibrations with a chosen section and fibers marked with a weight between 0 and 1. Moreover, we show that the domain of weights admits a wall and chamber structure, describe the induced wall-crossing morphisms between the moduli spaces as the weight vector varies, and describe the surfaces that appear on the boundary of the moduli space. The main technical result is a proof of invariance of log plurigenera for slc elliptic surface pairs with arbitrary weights. There are many technical obstacles in the theory of stable pairs, some of which are not completely resolved, and we expect the techniques developed here to overcome them will be useful in other applications.
Bejleri, Dori,
"A tale of two moduli spaces: Hilbert schemes of singular curves and moduli of elliptic surfaces"
(2018).
Mathematics Theses and Dissertations.
Brown Digital Repository. Brown University Library.
https://doi.org/10.26300/2w7s-5d71
