$\mathbb{Q}$-Fano threefolds with three birational Mori fiber structures

TL;DR

This paper constructs examples of $Q$-Fano threefolds with exactly three birational Mori fiber structures, revealing that this number can vary non-semicontinuously within families of such threefolds.

Contribution

It provides the first known examples of $Q$-Fano threefolds with precisely three birational Mori fiber structures, constructed as weighted hypersurfaces.

Findings

Examples of $Q$-Fano threefolds with three Mori fiber structures

Number of structures does not behave upper semi-continuously in families

Weighted hypersurfaces used for construction

Abstract

In this paper we give first examples of $\mathbb{Q}$-Fano threefolds whose birational Mori fiber structures consist of exactly three $\mathbb{Q}$-Fano threefolds. These examples are constructed as weighted hypersurfaces in a specific weighted projective space. We also observe that the number of birational Mori fiber structures does not behave upper semi-continuously in a family of $\mathbb{Q}$-Fano threefolds.