Stable totally degenerate hyperelliptic curves

Abstract

This paper deals with stable totally degenerate curves as well as disconnections (i.e. partial desingularizations) of such curves. By totally degenerate we mean that the normalization of each component is a projective line and each singularity is an ordinary double point. For an irreducible totally degenerate curve the existence of an involution is taken as a criterion for hyperellipticity. This is the starting-point in the first section from which we come by a natural compactification of the related moduli scheme to the notion of a properly hyperelliptic curve and the properly hyperelliptic locus H 2 g of the moduli space B 2 g of 2 g-pointed stable curves of genus 0. We give a characterization by involutions of the hyperelliptic curves constructed by Harris and Mumford in [HM]. Their definition is not identical, but close to ours. In the second section we describe H 2 g by equations and give a presentation as successive blow-up of H 2 g − 2 × P 1. This gives the possibility of calculating Chow groups and Betti numbers. We compute an explicit formula for the rank of the Picard group.