The Scorza correspondence in genus 3

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.