Covers of degree four and the rationality of the moduli space of curves of genus five with a vanishing theta-null

Abstract

Let [Mfr ]_g be the moduli space of smooth curves C of genus g[ges ]4 over the complexfield [Copf] and, as in [4], denote by [Mfr ]⁽ⁿ⁾_g⊂[Mfr ]_gthe locus of points representing curveshaving n vanishing theta-null, i.e. n points of order two in Sing (Θ).

I. V. Dolgachev proved the rationality of [Mfr ]⁽¹⁾₄(see [4, p. 13]) and of [Mfr ]⁽²⁾₅(see [4, p. 11]). The aim of this paper is to prove the

Theorem. [Mfr ]⁽¹⁾_tis rational.