Cubic equations for the hyperelliptic locus
Abstract
We discuss the conjecture of Buchstaber and Krichever that their multi-dimensional vector addition formula for Baker-Akhiezer functions characterizes Jacobians among principally polarized abelian varieties, and prove that it is indeed a weak characterization, i.e. that it is true up to additional components, or true precisely under a general position assumption. We also show that this addition formula is equivalent to Gunning's multisecant formula for the Kummer variety. We then use Buchstaber-Krichever's computation of the coefficients in the addition formula to obtain cubic relations among theta functions, which (weakly) characterize the locus of hyperelliptic Jacobians among irreducible abelian varieties. In genus 3 our equations are equivalent to the vanishing of one theta-null, and thus are known classically by work of Mumford and Poor, but already for genus 4 they appear to be new.
Cite
@article{arxiv.math/0503026,
title = {Cubic equations for the hyperelliptic locus},
author = {Samuel Grushevsky},
journal= {arXiv preprint arXiv:math/0503026},
year = {2007}
}