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Simultaneous semi-stable reduction for curves with ADE singularities

Authors: Sebastian Casalaina-Martin and Radu Laza
Journal: Trans. Amer. Math. Soc. 365 (2013), 2271-2295
MSC (2010): Primary 14H10; Secondary 14L24, 14E30
Published electronically: October 15, 2012
MathSciNet review: 3020098
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Abstract: A key tool in the study of algebraic surfaces and their moduli is Brieskorn's simultaneous resolution for families of algebraic surfaces with simple (du Val or ADE) singularities. In this paper we show that a similar statement holds for families of curves with at worst simple (ADE) singularities. For a family $ \mathscr X\to B$ of ADE curves, we give an explicit and natural resolution of the rational map $ B\dashrightarrow \overline M_g$. Moreover, we discuss a lifting of this map to the moduli stack $ \overline {\mathcal M}_g$, i.e. a simultaneous semi-stable reduction for the family $ \mathscr X/B$. In particular, we note that in contrast to what might be expected from the case of surfaces, the natural Weyl cover of $ B$ is not a sufficient base change for a lifting of the map $ B\dashrightarrow \overline M_g$ to $ \overline {\mathcal M}_g$.

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Additional Information

Sebastian Casalaina-Martin
Affiliation: Department of Mathematics, University of Colorado at Boulder, Boulder, Colorado 80309

Radu Laza
Affiliation: Department of Mathematics, Stony Brook University, Stony Brook, New York 11794

Received by editor(s): February 18, 2011
Published electronically: October 15, 2012
Additional Notes: The second author was partially supported by NSF grant DMS-0968968
Article copyright: © Copyright 2012 American Mathematical Society

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