Contractions and deformations

Donovan, W. and Wemyss, M. (2019) Contractions and deformations. American Journal of Mathematics, 141(3), pp. 563-592. (doi:10.1353/ajm.2019.0018)

Donovan, W. and Wemyss, M. (2019) Contractions and deformations. American Journal of Mathematics, 141(3), pp. 563-592. (doi:10.1353/ajm.2019.0018)

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Abstract

Suppose that f is a projective birational morphism with at most one-dimensional fibres between d-dimensional varieties X and Y , satisfying Rf∗OX = OY . Consider the locus L in Y over which f is not an isomorphism. Taking the scheme-theoretic fibre C over any closed point of L, we construct algebras Afib and Acon which prorepresent the functors of commutative deformations of C, and noncommutative deformations of the reduced fibre, respectively. Our main theorem is that the algebras Acon recover L, and in general the commutative deformations of neither C nor the reduced fibre can do this. As the d = 3 special case, this proves the following contraction theorem: in a neighbourhood of the point, the morphism f contracts a curve without contracting a divisor if and only if the functor of noncommutative deformations of the reduced fibre is representable.

Item Type:Articles
Additional Information:Research of the first author supported by World Premier International Research Center Initiative (WPI), MEXT, Japan, and by EPSRC grant EP/G007632/1; research of the second author supported by EPSRC grant EP/K021400/1.
Status:Published
Refereed:Yes
Glasgow Author(s) Enlighten ID:Wemyss, Professor Michael
Authors: Donovan, W., and Wemyss, M.
College/School:College of Science and Engineering > School of Mathematics and Statistics > Mathematics
Journal Name:American Journal of Mathematics
Publisher:John Hopkins University Press
ISSN:0002-9327
ISSN (Online):1080-6377
Copyright Holders:Copyright © 2019 Johns Hopkins University Press
First Published:First published in American Journal of Mathematics 141(3):563-592
Publisher Policy:Reproduced in accordance with the copyright policy of the publisher

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Project CodeAward NoProject NamePrincipal InvestigatorFunder's NameFunder RefLead Dept
624761Assessing the impact of foot-and-mouth disease vaccination programmesRichard ReeveBiotechnology and Biological Sciences Research Council (BBSRC)BB/K021400/1RI BIODIVERSITY ANIMAL HEALTH & COMPMED
3015810The Homological Minimal Model ProgramMichael WemyssEngineering and Physical Sciences Research Council (EPSRC)EP/R009325/1M&S - Mathematics