A new look at the moduli space of stable hyperelliptic curves

Abstract

We carry out the log minimal model program for the moduli space \({\bar H_g}\) of stable hyperelliptic curves and show that certain log canonical models of \({\bar H_g}\) are isomorphic to the proper transform of \({\bar H_g}\) in the corresponding log canonical models of \({\bar M_g}\). For g = 3, we retrieve the compact moduli space \({\bar B_{8}}\) of binary forms as a log canonical model, and obtain a decomposition of the natural map \({\bar H_3 \to \bar B_{8}}\) into successive divisorial contractions of the boundary divisors. As a byproduct, we also obtain an isomorphism of \({\bar B_8}\) with the GIT quotient of the Chow variety of bicanonically embedded hyperelliptic curves of genus three.