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Jacobians with a vanishing theta-null in genus 4

Samuel Grushevsky·Riccardo Salvati Manni
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Abstract

In this paper we prove a conjecture of Hershel Farkas that if a 4-dimensional principally polarized abelian variety has a vanishing theta-null, and the hessian of the theta function at the corresponding point of order two is degenerate, the abelian variety is a Jacobian. We also discuss possible generalizations to higher genera, and an interpretation of this condition as an infinitesimal version of Andreotti and Mayer's local characterization of Jacobians by the dimension of the singular locus of the theta divisor.

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Samuel Grushevsky, Riccardo Salvati Manni. 2006-05-06. Jacobians with a vanishing theta-null in genus 4. https://arxiv.org/abs/math/0605160

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