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Gradients of odd theta functions

Samuel Grushevsky·Riccardo Salvati Manni
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Abstract

We show that a generic principally polarized abelian variety (ppav) is uniquely determined by its theta hyperplanes. These are the non-projectivized version of those studied by Caporaso and Sernesi (see math.AG/0204164), which in a sense are a generalization to ppavs of bitangents of plane quartics, and of hyperplanes tangent to a canonical curves of genus $g$ in $g-1$ points. 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.

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Samuel Grushevsky, Riccardo Salvati Manni. 2003-10-07. Gradients of odd theta functions. https://arxiv.org/abs/math/0310085

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