Monodromy of stable curves of compact type: rigidity and extension

Abstract

Let ${\cal M}_{g,n}$, for $2g-2+n>0$, be the moduli stack of $n$-pointed, genus $g$, smooth curves. For a family $C\to S$ of such curves over a connected base and a geometric point $\xi$ on $S$, the associated monodromy representation is the induced homomorphism $\pi_1(S,\xi)\to\pi_1({\cal M}_{g,n},[C_\xi])$ on algebraic fundamental groups. It is well known that, if $S$ is irreducible, reduced and locally of finite type over a field $k$ of characteristic zero, the fibre $C_\xi$ and the corresponding monodromy representation determine the relative isomorphism class of the family. In the first part of the paper, it is shown that suitable quotients of this representation suffice. These results are then applied to show that the monodromy representation associated to a family $C\to S$ of $n$-pointed, genus $g$, stable curves of compact type, i.e. the induced homomorphism $\pi_1(S,\xi)\to\pi_1(\widetilde{\cal M}_{g,n},[C_\xi])$ (where, $\widetilde{\cal M}_{g,n}$ denotes the moduli stack of $n$-pointed, genus $g$, stable curves of compact type), characterizes trivial and isotrivial families. Let $U$ be an open subscheme of a normal, irreducible, locally noetherian scheme $S$ over a field $k$ of characteristic zero and let $C\ra U$ be a family of stable curves of compact type. In the second part of the paper, a monodromy criterion is given for extending $C\to U$ to a family of stable curves of compact type over $S$.