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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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