Any smooth plane quartic can be reconstructed from its bitangents

Abstract

In this paper, we present two related results on curves of genus 3. The first gives a bijection between the classes of the following objects: * Smooth non-hyperelliptic curves C of genus 3, with a choice of an element a in Jac(C)[2]-{0}, such that the cover C/|K_C+a|^* does not have an intermediate factor; up to isomorphism. * Plane curves E,Q in P^2 and an element in b' in Pic(E)[2]-{0}, where E,Q are of degrees 3,2, the curve E is smooth and Q,E intersect transversally ; up to projective transformations. We discuss the degenerations of this bijection, and give an interpretation of the bijection in terms of Abelian varieties. Next, we give an application of this correspondence: An EXPLICIT proof of the reconstructability of ANY smooth plane quartic from its bitangents.