The Rationality of Certain Moduli Spaces of Curves of Genus 3 Ingrid Bauer and Fabrizio Catanese

The aim of this paper is to give an explicit geometric description of the birational structure of the moduli space of pairs (C, η), where C is a general curve of genus 3 over an algebraically closed field k of arbitrary characteristic and η ∈ Pic(C)3 is a non trivial divisor class of 3-torsion on C. As it was observed in [B-C04] lemma (2.18), if C is a general curve of genus 3 and η ∈ Pic(C)3 is a non trivial 3 torsion divisor class, then we have a morphism φη := φ|KC+η| × φ|KC−η| : C → P 1 × P, corresponding to the sum of the linear systems |KC + η| and |KC − η|, which is birational onto a curve Γ ⊂ P × P of bidegree (4, 4). Moreover, Γ has exactly six ordinary double points as singularities, located in the six points of the set S := {(x, y)|x 6= y, x, y ∈ {0, 1,∞}}. In [B-C04] we only gave an outline of the proof (and there is also a minor inaccuracy). Therefore we dedicate the first section of this article to a detailed geometrical description of such pairs (C, η), where C is a general curve of genus 3 and η ∈ Pic(C)3 \ {0}. The main result of the first section is the following: