A family of proper smooth curves of genus $\geq 2$, parametrised by an open
dense subset $U$ of a normal variety $S$, extends to $S$ if the natural map
$\pi_1(U) \to \pi_1(S)$ on fundamental groups is an isomorphism. The criterion
of this note is actually more precise.Comment: 12 pages; revised version including a section on outer
  representations for families of curve

Stix, Jakob

English

arXiv

Abstract — A family of proper smooth curves of genus ≥ 2, parametrised by an open dense subset U of a normal variety S, extends to S if the natural map pi1U → pi1S is an isomorphism. The criterion of this note is actually more precise. 1 The monodromy criterion (1.1) All schemes are locally noetherian. For a scheme S we denote by Mg,n(S) the groupoid of smooth, proper S-curves with geometrically connected fibres of genus g and n disjoint ordered sections, see [Kn83]. If 2 − 2g − n is negative, Mg,n is a smooth Deligne-Mumford stack over Spec(Z). The main result of this note is the following theorem. Theorem 1.2 (monodromy criterion I). Let 2 − 2g−n be negative. For a dense open subscheme U of a normal, excellent scheme S the following holds. (1) The restriction functor Mg,n(S)→Mg,n(U) is fully faithful. (2) A U-curve C ∈ Mg,n(U) extends to an S-curve in Mg,n(S) if and only if for all N ∈ N and geometric points u ∈ U⊗Z [ 1N] the following commutative diagram exists, where the solid arrows are induced by the natural maps U →Mg,n and U ⊂ S: pi1(U ⊗ Z [ 1N], u) // ((RR RR

Jakob Stix

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A monodromy criterion for extending curves

http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.241.6076

A monodromy criterion for extending curves

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