Addition formulae, multisecants, and the hyperelliptic locus

Sam Grushevsky
Princeton U., USA
 
 

Abstract

---

We show that the multi-dimensional vector addition formula for Baker-Akhiezer functions obtained by Krichever and Buchstaber is equivalent to existence of multisecants for the Kummer variety, as described by Gunning, and that it characterizes Jacobians among principally polarized abelian varieties.

We then use this formula to obtain cubic relations among theta functions of hyperelliptic Jacobians, characterizing the hyperelliptic locus precisely. In genus 3 our equations are equivalent to the vanishing of a theta-null, and thus are classical, but already for genus 4 they appear to be new.