Radu Laza

The moduli space of cubic threefolds via degenerations of the intermediate Jacobian

Abstract A well known result of Clemens and Griffiths says that a smooth cubic threefold can be recovered from its intermediate Jacobian. In this paper we discuss the possible degenerations of these abelian varieties, and thus give a description of the compactification of the moduli space of cubic threefolds obtained in this way. The relation between this compactification and those constructed in the work of Allcock-Carlson-Toledo and Looijenga-Swierstra is also considered, and is similar in spirit to the relation between the various compactifications of the moduli spaces of low genus curves.

Log canonical models and variation of GIT for genus 4 canonical curves

We discuss GIT for canonically embedded genus four curves and the connection to the Hassett–Keel program. A canonical genus four curve is a complete intersection of a quadric and a cubic, and, in contrast to the genus three case, there is a family of GIT quotients that depend on a choice of linearization. We discuss the corresponding VGIT problem and show that the resulting spaces give the final steps in the Hassett–Keel program for genus four curves.

Simultaneous semi-stable reduction for curves with ADE singularities

A key tool in the study of algebraic surfaces and their moduli is Brieskorns simultaneous resolution for families of algebraic surfaces with simple (du Val or ADE) singularities. In this paper we show that a similar statement holds for families of curves with at worst simple (ADE) singularities. For a family

Perspectives on the Construction and Compactification of Moduli Spaces

\mathscr X\to B

Classical Period Domains

of ADE curves, we give an explicit and natural resolution of the rational map

Extending the Prym map to toroidal compactifications of the moduli space of abelian varieties (with an appendix by Mathieu Dutour Sikirić)

B\to \bar M_g

Counting the hyperplane sections with fixed invariants of a plane quintic – three approaches to a classical enumerative problem

. Moreover, we discuss a lifting of this map to the moduli stack

The moduli space of cubic fourfolds via the period map

\bar {\mathcal M}_g

The KSBA compactification for the moduli space of degree two

, i.e. a simultaneous semi-stable reduction for the family

K

\mathscr X/B