One-dimensional substacks of the moduli stack of Deligne-Mumford stable curves

Details One-dimensional substacks of the moduli stack of Deligne-Mumford stable curves One-dimensional substacks of the moduli stack of Deligne-Mumford stable curves

2006-12-27

arxiv.org/abs/math/0612802v1

Mathematics

Algebraic Geometry

288 pages

Scientific paper

There is a well-known stratification of the moduli space $M_g$ of Deligne-Mumford stable curves of genus $g$ by the loci of stable curves with a fixed number $i$ of nodes, where $i \le 3g-3$. The associated moduli stack ${\cal M}_g$ admits an analogous stratification. Our main objects of study are those one-dimensional substacks of the moduli stack, which are the irreducible components of the stratum corresponding to stable curves with exactly $3g-4$ nodes. We describe these substacks explicitely as quotient stacks, and relate them to other, and simpler, moduli stacks of (permutation classes of) pointed stable curves. In an appendix, an extensive compendium on quotient stacks is provided.

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