Gradients of odd theta functions

We show that a generic principally polarized abelian variety is uniquely determined by its gradient theta-hyperplanes, the non-projectivized version of those studied in [1], [2], [3], which in a sense are a generalization to ppavs of bitangents of plane curves. More precisely, we show that, generically, the set of gradients of all odd theta functions at the point zero uniquely determines a ppav with level (4,8) structure. We also show that our map is an immersion of the moduli space of ppavs.