Copyright © The Author 2007. Published by Oxford University Press.
Monodromy of Stable Curves of Compact Type: Rigidity and Extension
Institut de Mathématiques de Jussieu; 175, rue du Chevaleret; 75013 Paris
Correspondence: Correspondence to be sent to: Institut de Mathématiques de Jussieu; 175, rue du Chevaleret; 75013 Paris. boggi{at}math.jussieu.fr
Let 
g , n, for 2g – 2 + n > 0, be the moduli stack of n-pointed, genus g, smooth curves. For a family C 
S of such curves over a connected base and a geometric point 
on S, the associated monodromy representation is the induced homomorphism 
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 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 S of n-pointed, genus g, stable curves of compact type, that is, the induced homomorphism 
(where, 
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 S be a family of stable curves of compact type. In the scond part of he paper, a monodromy criterion is given for extending C S to a family of stable curves of compact type over S.