Deforming elephants of Q-Fano threefolds

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.

31 pages, to appear in Jounal of LMS

References in corpus (1)

• On G-Fano threefolds

Cited by in corpus (2)

• On deformations of Q-Fano threefolds II
• Deformations of weak $\mathbb{Q}$-Fano 3-folds