High-pliability Fano hypersurfaces

TL;DR

This paper demonstrates that certain Fano 3-fold hypersurfaces with specific singularities have multiple distinct birational models, revealing their pliability and complex birational geometry.

Contribution

It identifies five Fano 3-fold hypersurfaces with compound Du Val singularities that have pliability at least two, showing they admit at least two non-isomorphic birational models.

Findings

Five Fano 3-fold hypersurfaces have pliability ≥ 2.

The birational map is composed of two links from blowing up Type I centers.

The models differ by their singularity types and embeddings.

Abstract

We show that five of Reid's Fano 3-fold hyperurfaces containing at least one compound Du Val singularity of type $cA_n$ have pliability at least two. The two elements of the pliability set are the singular hypersurface itself, and another non-isomorphic Fano hypersurface of the same degree, embedded in the same weighted projective space, but with different compound Du Val singularities. The birational map between them is the composition of two birational links initiated by blowing up two Type I centres on a codimension 4 Fano 3-fold of $\mathbb{P}^2 \times \mathbb{P}^2$-type having Picard rank 2.