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 |
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| 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 |
| Related URLs: |
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