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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Deformations of singularities and variation of GIT quotients


Author: Radu Laza
Journal: Trans. Amer. Math. Soc. 361 (2009), 2109-2161
MSC (2000): Primary 14J17, 14B07, 32S25; Secondary 14L24
Published electronically: November 12, 2008
MathSciNet review: 2465831
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Abstract: We study the deformations of the minimally elliptic surface singularity $ N_{16}$. A standard argument reduces the study of the deformations of $ N_{16}$ to the study of the moduli space of pairs $ (C,L)$ consisting of a plane quintic curve and a line. We construct this moduli space in two ways: via the periods of $ K3$ surfaces and by using geometric invariant theory (GIT). The GIT construction depends on the choice of the linearization. In particular, for one choice of linearization we recover the space constructed via $ K3$ surfaces and for another we obtain the full deformation space of $ N_{16}$. The two spaces are related by a series of explicit flips. In conclusion, by using the flexibility given by GIT and the standard tools of Hodge theory, we obtain a good understanding of the deformations of $ N_{16}$.


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Additional Information

Radu Laza
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
Email: rlaza@umich.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-08-04660-6
PII: S 0002-9947(08)04660-6
Received by editor(s): August 28, 2006
Received by editor(s) in revised form: May 21, 2007
Published electronically: November 12, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.