On deformations of Q-Fano threefolds

TL;DR

This paper investigates the deformation properties of Q-Fano threefolds with terminal singularities, establishing smoothness of their deformation space and conditions for smoothability to simpler quotient singularities, with applications to genus bounds.

Contribution

It proves the smoothness of the Kuranishi space for Q-Fano 3-folds and introduces conditions under which they are Q-smoothable, advancing understanding of their deformation theory.

Findings

Kuranishi space of Q-Fano 3-folds is smooth.

Q-Fano 3-folds with ordinary terminal singularities are Q-smoothable.

Q-smoothability is achieved under the existence of a Du Val anticanonical element.

Abstract

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