Deforming elephants of Q-Fano threefolds

TL;DR

This paper investigates deformations of $Q$-Fano 3-folds with terminal singularities and their elephants, showing conditions under which they can be deformed to simpler singularity types and providing examples of more complex cases.

Contribution

It introduces new deformation techniques for $Q$-Fano 3-folds and their elephants, especially when elephants have non-normal singularities, expanding understanding of their deformation behavior.

Findings

Deformation to quotient singularities and Du Val elephants is possible under certain conditions.

Reduces the existence problem of deformations to a local problem around singularities.

Provides examples of $Q$-Fano 3-folds lacking Du Val elephants.

Abstract

We study deformations of a pair of a $\mathbb{Q}$-Fano $3$-fold $X$ with only terminal singularities and its elephant $D \in |{-}K_X|$. We prove that, if there exists $D \in |{-}K_X|$ with only isolated singularities, the pair $(X,D)$ can be deformed to a pair of a $\mathbb{Q}$-Fano $3$-fold with only quotient singularities and a Du Val elephant. When there are only non-normal elephants, we reduce the existence problem of such a deformation to a local problem around the singular locus of the elephant. We also give several examples of $\mathbb{Q}$-Fano $3$-folds without Du Val elephants.