• admissible cover
• algebraic geometry
• curve
• moduli
• stack

I describe two possible compactifications of the stack of smooth hyperelliptic curves of genus g and discuss their relationships. The first one parametrizes stable hyperelliptic curves, that is, families of stable curves of genus g together with an involution with isolated fixed points and whose quotient is a nodal curve of genus 0. This may be obtained as the closure of the hyperelliptic locus inside the stack of stable curves. The second compactification is the so-called stack of admissible double covers. Its objects are families of nodal curves together with an involution with isolated fixed points, such that the quotient is a nodal curve of genus 0, stable when marked with the branch divisor of the étale locus, and such that the quotient map sends nodes to nodes. It turns out that there exists a natural map from the second stack to the first, given by collapsing the unstable components. This does not give an isomorphism of stacks. However, it is bijective on geometric points, proper and its ramification may be explicitly computed. This allows us to prove that, in characteristic zero, the induced map between coarse moduli spaces is indeed an isomorphism.