The moduli space of curves of genus three together with an odd theta-characteristic is rational

Abstract

In this note we use a “normal form”, due to Sylvester, for the equation of a generic cubic surface in ℙ³(ℂ) to prove that %plane1D;4B0;= {moduli space of pairs (S,P) with S smooth cubic surface, P a point on S} is rational. We then prove that %plane1D;510;₃^oth = {moduli space of curves of genus three together with one odd theta-characteristic} is birational to %plane1D;4B0; and so rational.