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Non-normal abelian covers

Part of: Surfaces and higher-dimensional varieties Birational geometry

Published online by Cambridge University Press:  20 March 2012

Valery Alexeev  and
Rita Pardini
Valery Alexeev
Affiliation:
Department of Mathematics, University of Georgia, Athens, GA 30605, USA (email: [email protected])
Rita Pardini
Affiliation:
Dipartimento di Matematica, Università di Pisa, Largo B. Pontecorvo 5, 56127 Pisa, Italy (email: [email protected])
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Abstract

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An abelian cover is a finite morphism X→Y of varieties which is the quotient map for a generically faithful action of a finite abelian group G. Abelian covers with Y smooth and X normal were studied in [R. Pardini, Abelian covers of algebraic varieties, J. Reine Angew. Math. 417 (1991), 191–213; MR 1103912(92g:14012)]. Here we study the non-normal case, assuming that X and Y are S2 varieties that have at worst normal crossings outside a subset of codimension greater than or equal to two. Special attention is paid to the case of ℤr2-covers of surfaces, which is used in [V. Alexeev and R. Pardini, Explicit compactifications of moduli spaces of Campedelli and Burniat surfaces, Preprint (2009), math.AG/arXiv:0901.4431] to construct explicitly compactifications of some components of the moduli space of surfaces of general type.

Keywords

coveringsabelian coversnon-normal varietiessurface singularitiesstable surfaces

MSC classification

Secondary: 14E20: Coverings 14J17: Singularities
Type
Research Article
Information
Compositio Mathematica , Volume 148 , Issue 4 , July 2012 , pp. 1051 - 1084
Copyright
Copyright © Foundation Compositio Mathematica 2012

References

[AP09]Alexeev, V. and Pardini, R., Explicit compactifications of moduli spaces of Campedelli and Burniat surfaces, Preprint (2009), math.AG/arXiv:0901.4431.Google Scholar
[AK70]Altman, A. and Kleiman, S., Introduction to Grothendieck duality theory, Lecture Notes in Mathematics, vol. 146 (Springer, Berlin, 1970); MR 0274461(43#224).CrossRefGoogle Scholar
[Bou65]Bourbaki, N., Éléments de mathématique. Fasc. XXXI, in Algèbre commutative chapitre 7: diviseurs, Actualités Scientifiques et Industrielles, vol. 1314 (Hermann, Paris, 1965); MR 0260715(41#5339).Google Scholar
[FP97]Fantechi, B. and Pardini, R., Automorphisms and moduli spaces of varieties with ample canonical class via deformations of abelian covers, Comm. Algebra 25 (1997), 1413–1441; MR 1444010(98c:14028).CrossRefGoogle Scholar
[Gro65]Grothendieck, A., Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. II, Publ. Math. Inst. Hautes Études Sci. 24 (1965), MR 0199181(33#7330).Google Scholar
[Har67]Hartshorne, R., Local cohomology: a seminar given by A. Grothendieck, Harvard University, Fall, 1961, Lecture Notes in Mathematics, vol. 41 (Springer, Berlin, 1967); MR 0224620(37#219).CrossRefGoogle Scholar
[Iac06]Iacono, D., Local structure of abelian covers, J. Algebra 301 (2006), 601–615; MR 2236759(2007d:14034).CrossRefGoogle Scholar
[KS88]Kollár, J. and Shepherd-Barron, N. I., Threefolds and deformations of surface singularities, Invent. Math. 91 (1988), 299–338; MR 922803(88m:14022).CrossRefGoogle Scholar
[Par91]Pardini, R., Abelian covers of algebraic varieties, J. Reine Angew. Math. 417 (1991), 191–213; MR 1103912(92g:14012).Google Scholar
[Rei80]Reid, M., Canonical 3-folds, in Journées de Géometrie Algébrique d’Angers, Juillet 1979 (Algebraic Geometry, Angers, 1979) (Sijthoff & Noordhoff, Alphen aan den Rijn, 1980), 273–310; MR 605348(82i:14025).Google Scholar