The Scorza correspondence in genus 3

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, σ(x) {pipe} x ∈ C} (where σ is the hyperelliptic involution), and twice the diagonal. Our proof uses the geometry of the subsystem Γ₀₀ of the linear system {pipe}2Θ{pipe}, and Riemann identities for theta constants.