The moduli spaces of hyperelliptic curves and binary forms

Abstract

There is a canonical isomorphism between the coarse moduli spaces of somooth hyperelliptic curves of genus g and binary forms of degree 2g+2 with nonzero discriminant. In this paper, we study the extension of this isomorphism to the compactification of these moduli spaces. The main result is that the canonical isomorphism above extends to a holomorphic map f_g from the compactication of the moduli space of hyperelliptic curves of genus g to the compactification of the moduli space of binary forms of degree 2g+2. Moreover, we work out how the boundary components are contracted by the map f_g.