A new look at the moduli space of stable hyperelliptic curves

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