The Scorza correspondence in genus 3

TL;DR

This paper proves a specific case of a conjecture relating to the limit of the Scorza correspondence for genus 3 hyperelliptic curves, using theta function identities and geometric analysis.

Contribution

It establishes the genus 3 case of Farkas and Verra's conjecture on the Scorza correspondence limit for curves with a theta-null.

Findings

The limit of the Scorza correspondence is the union of the hyperelliptic involution curve and twice the diagonal.

The proof employs the geometry of the a0Gamma_{00} subsystem of 2\u0398 and Riemann identities.

Confirmed the conjecture for genus 3 hyperelliptic curves.

Abstract

In this note we prove the genus 3 case of a conjecture of G. Farkas and A. 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 ${x,\sigma(x))$ (where $\sigma$ is the hyperelliptic involution), and twice the diagonal. Our proof uses the geometry of the subsystem \Gamma_{00} of the linear system 2\Theta, and Riemann identities for theta constants.