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The Scorza correspondence in genus 3

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Abstract

In this note we prove the genus 3 case of a conjecture of Farkas and Verra on the limit of the Scorza correspondence for curves with a theta-null. Specifically, we show that the limit of the Scorza correspondence for a hyperelliptic genus 3 curve C is the union of the curve \({\{x, \sigma(x) \mid x \in C\}}\) (where σ is the hyperelliptic involution), and twice the diagonal. Our proof uses the geometry of the subsystem Γ00 of the linear system |2Θ|, and Riemann identities for theta constants.

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Authors and Affiliations

  1. Mathematics Department, Stony Brook University, Stony Brook, NY, 11790-3651, USA

    Samuel Grushevsky

  2. Dipartimento di Matematica, Università “La Sapienza”, Piazzale A. Moro 2, 00185, Roma, 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. The Scorza correspondence in genus 3. manuscripta math. 141, 111–124 (2013). https://doi.org/10.1007/s00229-012-0564-z

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  • DOI: https://doi.org/10.1007/s00229-012-0564-z

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