Simultaneous semi-stable reduction for curves with ADE singularities
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- by Sebastian Casalaina-Martin and Radu Laza PDF
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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 $\overline {\mathcal M}_g$.References
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Additional Information
- Sebastian Casalaina-Martin
- Affiliation: Department of Mathematics, University of Colorado at Boulder, Boulder, Colorado 80309
- MR Author ID: 754836
- Email: casa@math.colorado.edu
- Radu Laza
- Affiliation: Department of Mathematics, Stony Brook University, Stony Brook, New York 11794
- MR Author ID: 692317
- ORCID: 0000-0001-9631-1361
- Email: rlaza@math.sunysb.edu
- 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
- © Copyright 2012 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 365 (2013), 2271-2295
- MSC (2010): Primary 14H10; Secondary 14L24, 14E30
- DOI: https://doi.org/10.1090/S0002-9947-2012-05579-6
- MathSciNet review: 3020098