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
DOI: https://doi.org/10.1090/jag/672
Published electronically: August 4, 2015
MathSciNet review: 3419958
Full-text PDF

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.


References [Enhancements On Off] (What's this?)


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
Email: tarosano222@gmail.com

DOI: https://doi.org/10.1090/jag/672
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.

Journal of Algebraic Geometry
The Journal of Algebraic Geometry
is sponsored by the Department of Mathematical Sciences
of Tsinghua University
and is distributed by the American Mathematical Society
for University Press, Inc.
Online ISSN 1534-7486; Print ISSN 1056-3911
© 2017 University Press, Inc.
Comments: jag-query@ams.org
AMS Website