On the Coble quartic

We review and extend the known constructions relating Kummer threefolds, Gopel systems, theta constants and their derivatives, and the GIT quotient for 7 points in P^2 to obtain an explicit expression for the Coble quartic. The Coble quartic was recently determined completely by Q.Ran, S.Sam, G.Schrader, and B.Sturmfels, who computed it completely explicitly, as a polynomial with 372060 monomials of bidegree (28,4) in theta constants of the second order and theta functions of the second order, respectively. Our expression is in terms of products of theta constants with characteristics corresponding to Gopel systems, and is a polynomial with 134 terms. Our approach is based on the beautiful geometry studied by Coble and further investigated by Dolgachev and Ortland, and highlights the geometry and combinatorics of syzygetic octets of characteristics, and the corresponding representations of the symplectic group. One new ingredient is the relationship of Gopel systems and Jacobian determinants of theta functions. In genus 2, we similarly obtain a short explicit equation for the universal Kummer surface, and relate modular forms of level two to binary invariants of six points on P^1

On the Coble quartic

We review and extend the known constructions relating Kummer threefolds, G¨opel systems, theta constants and their derivatives, and the GIT quotient for 7 points in P^2 to obtain an explicit expression for the Coble quartic. The Coble quartic was recently determined completely in [RSSS12], where it was computed completely explicitly, as a polynomial with 372060 monomials of bidegree (28, 4) in theta constants of the second order and theta functions of the second order, respectively. Our expression is in terms of products of theta constants with characteristics corresponding to G¨opel systems, and is a polynomial with 134 terms. Our approach is based on the beautiful geometry studied by Coble and further investigated by Dolgachev and Ortland and highlights the geometry and combinatorics of syzygetic octets of characteristics, and the corresponding representations of Sp(6, F_2). One new ingredient is the relationship of G¨opel systems and Jacobian determinants of theta functions. In genus 2, we similarly obtain a short explicit equation for the universal Kummer surface, and relate modular forms of level two to binary invariants of six points on P^

On singular Luroth quartics

Plane quartics containing the ten vertices of a complete pentalateral and limits of them are called L\"uroth quartics. The locus of singular L\"uroth quartics has two irreducible components, both of codimension two in $\P^{14}$. We compute the degree of them and we discuss the consequences of this computation on the explicit form of the L\"uroth invariant. One important tool are the Cremona hexahedral equations of the cubic surface. We also compute the class in $\overline{M}_3$ of the closure of the locus of nonsingular L\"uroth quartics.Comment: Enlarged version. Computation of the class of the locus of Luroth quartics in the moduli space adde

Orbit Parametrizations for K3 Surfaces

We study moduli spaces of lattice-polarized K3 surfaces in terms of orbits of representations of algebraic groups. In particular, over an algebraically closed field of characteristic 0, we show that in many cases, the nondegenerate orbits of a representation are in bijection with K3 surfaces (up to suitable equivalence) whose N\'eron-Severi lattice contains a given lattice. An immediate consequence is that the corresponding moduli spaces of these lattice-polarized K3 surfaces are all unirational. Our constructions also produce many fixed-point-free automorphisms of positive entropy on K3 surfaces in various families associated to these representations, giving a natural extension of recent work of Oguiso.Comment: 83 pages; to appear in Forum of Mathematics, Sigm

Moduli of Abelian varieties, Vinberg theta-groups, and free resolutions

We present a systematic approach to studying the geometric aspects of Vinberg theta-representations. The main idea is to use the Borel-Weil construction for representations of reductive groups as sections of homogeneous bundles on homogeneous spaces, and then to study degeneracy loci of these vector bundles. Our main technical tool is to use free resolutions as an "enhanced" version of degeneracy loci formulas. We illustrate our approach on several examples and show how they are connected to moduli spaces of Abelian varieties. To make the article accessible to both algebraists and geometers, we also include background material on free resolutions and representation theory.Comment: 41 pages, uses tabmac.sty, Dedicated to David Eisenbud on the occasion of his 65th birthday; v2: fixed some typos and added reference