Nodal cubic surfaces and the rationality of the moduli space of curves of genus two (Q794736)

Let \({\mathcal M}_ 2\) be the moduli space of curves of genus 2 and \({\mathcal S}\) be the moduli space of ''nodal cubics'' i.e. of cubic surfaces with one ordinary double point. We produce a birational map \({\mathcal S}\to {\mathcal M}_ 2\) as follows: to a nodal cubic S with note P we associate a genus 2 curve \(\Gamma_ S\) which is the double cover of the conic \({\mathbb{P}}roj(\tan gent\quad cone\quad to\quad S\quad at\quad P)\) branched at the six points which are the lines through P lying on S; conversely, the bicanonical image of a genus 2 curve \(\Gamma\) is a conic in \({\mathbb{P}}^ 2\) with six distinguished points on it and this determines a nodal cubic. By using the classical Sylvester's theorem which gives a unique ''canonical'' equation for a generic (even nodal) cubic surface we construct an explicit ''birational model'' W for \({\mathcal S}\). W is a hypersurface of \({\mathbb{P}}^ 4/\sigma_ 5\) where \(\sigma_ 5\) is the symmetric group on 5 letters. The rationality of W is then proved by some computations, thus yielding the rationality of \({\mathcal S}\) and \({\mathcal M}_ 2\) over an algebraically closed field of characteristic 0.