2012/12/09 by Samuel Grushevsky, Grushevsky, Samuel, Riccardo Salvati Manni +1
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.1212.1895
openalex publication_date 2012/12/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 P2 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 P1.