A new look at the moduli space of stable hyperelliptic curves

Abstract

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