The hyperelliptic locus with special reference to characteristic two

Abstract

Let Jg0 be the coarse moduli space for curves of genus g(>2) defined over an algebraically closed field k, and denote by o~'.0 the subset of ~¢/0 corresponding to the hyperelliptic curves (the hyperelliptic locus). It is proved below that Yfg is closed and irreducible, and that any two points of ~ may be connected by a (non-complete) rational curve reside ~ . In the case k= C this has been proved by the Italian geometers, see e.g. [9]; more recently Rauch [8] has proved the closedness of ~ , and ArbareUo [1] has shown that ~g is irreducible and unirational. For g= 2 and char(k) arbitrary, the results follow immediately from Igusa's explicit description of ~2 =~/g2, [5]. For arbitrary g, but char (k) :t: 2, the irreducibility of "~¢'0 is a trivial consequence of Geyer's construction of a factorial affine moduli space for the hypereltiptic curves of genus g, [4]. Our proof of the irreducibility and the (uni-)rationatity question follow the classical (Italian) pattern with a slight modification inspired by Igusa's treatment for genus two: we exhibit a rational family of plane curves of degree g + 2, possessing only one singularity, which is ordinary of multiplicity g, such that every birational isomorphism class of hyperelliptic curves is represented in the family. Next, we blow-up these singularities simultaneously and acquire a smooth family with analogous properties. The universal property of ~'g now yields the desired results, but the closedness of ~ . The latter is achieved by a cohomological argument. We should like to express our thanks to F. Oort, who proposed the irreducibility question in characteristic two.