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Jacobians with a vanishing theta-null in genus 4

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Abstract

In this paper we prove a conjecture of Hershel Farkas [11] that if a 4-dimensional principally polarized abelian variety has a vanishing theta-null, and the Hessian of the theta function at the corresponding 2-torsion point 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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Authors and Affiliations

  1. Mathematics Department, Princeton University, Fine Hall Washington Road, Princeton, NJ, 08544, USA

    Samuel Grushevsky

  2. Dipartimento di Matematica, Università “La Sapienza”, Piazzale A. Moro 2, Roma, I 00185, Italy

    Riccardo Salvati Manni

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  1. Samuel Grushevsky
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  2. Riccardo Salvati Manni
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Correspondence to Samuel Grushevsky.

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Grushevsky, S., Salvati Manni, R. Jacobians with a vanishing theta-null in genus 4. Isr. J. Math. 164, 303–315 (2008). https://doi.org/10.1007/s11856-008-0031-4

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  • DOI: https://doi.org/10.1007/s11856-008-0031-4

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