Hyperelliptic Gorenstein curves (Q1295683)

The subject of this paper is the moduli space of pointed singular hyperelliptic curves of genus \(g\geq 2\). It is proved that every such curve can be realized as a curve of degree \(2g+2\) on a rational cone in \({\mathbb P}^{g+2}\), not passing through its vertex, its \(g^1_2\) being cut on the curve by the lines of the cone. Moreover every morphism between two such curves is induced by an automorphism of \({\mathbb P}^{g+2}\) itself, which carries the cone to itself. Using this idea, it is proved that the classes of isomorphism of such curves possessing a two-branched point of singularity degree \(\delta \) form a family of dimension \(2g-2\delta\), while the family of those equipped with an unibranched Weierstrass point has dimension \(2g-2\delta -1\) (the dimension of the family with a non-Weierstrass point has dimension \(2g\)). The result is first proved in any characteristic \(p\neq 2\); the case \(p=2\) is then explicitly described.