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
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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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