Any Smooth Plane Quartic Can Be Reconstructed from Its Bitangents

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 α ∈ Jac(C)[2]r {0}, such that the cover C −→ |KC +α| ∗ does not have an intermediate factor; up to isomorphism. • Plane curves E,Q →֒ P and an element in β′ ∈ Pic(E)[2] r {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.