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Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



On deformations of $\mathbb {Q}$-Fano $3$-folds

Author: Taro Sano
Journal: J. Algebraic Geom. 25 (2016), 141-176
Published electronically: August 4, 2015
MathSciNet review: 3419958
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Abstract | References | Additional Information

Abstract: We study the deformation theory of a $\mathbb {Q}$-Fano 3-fold with only terminal singularities. First, we show that the Kuranishi space of a $\mathbb {Q}$-Fano 3-fold is smooth. Second, we show that every $\mathbb {Q}$-Fano 3-fold with only “ordinary” terminal singularities is $\mathbb {Q}$-smoothable; that is, it can be deformed to a $\mathbb {Q}$-Fano 3-fold with only quotient singularities. Finally, we prove $\mathbb {Q}$-smoothability of a $\mathbb {Q}$-Fano 3-fold assuming the existence of a Du Val anticanonical element. As an application, we get the genus bound for primary $\mathbb {Q}$-Fano 3-folds with Du Val anticanonical elements.

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Additional Information

Taro Sano
Affiliation: Mathematics Institute, Zeeman Building, University of Warwick, Coventry, CV4 7AL, United Kingdom – and – Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany

Received by editor(s): November 25, 2012
Received by editor(s) in revised form: July 15, 2013, March 14, 2014, March 21, 2014, August 5, 2014, and September 2, 2014
Published electronically: August 4, 2015
Additional Notes: The author was partially supported by a Warwick Postgraduate Research Scholarship. He was partially funded by the Korean government WCU Grant R33-2008-000-10101-0, Research Institute for Mathematical Sciences and Higher School of Economics
Dedicated: Dedicated to Professor Yujiro Kawamata on the occasion of his 60th birthday
Article copyright: © Copyright 2015 University Press, Inc.