On deformations of ℚ-Fano 3-folds

Abstract

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