The universal Kummer variety

Riccardo Salvati Manni (Roma Sapienza)

The Kummer surface is the quotient of an abelian surface by the involution sending x to –x. Using classical theta function theory, it can be represented as a quartic in P^3 whose coefficients are polynomials of degree 12 in the second order theta constants. Such a polynomial of (bi)degree (12,4) is an equation for the universal Kummer surface. Using several recent results, we intend to explain as one can extend this method to higher genera. In particular we will discuss Coble’s hypersurface in genus 3 and how one can obtain explicit equations for the Jacobian locus in the Siegel space.