$\mathbb{Q}$-Fano threefolds of Fano index thirteen

TL;DR

This paper classifies certain non-toric $Q$-factorial terminal Fano threefolds with Fano index 13, showing they are weighted hypersurfaces of degree 12 in a specific weighted projective space.

Contribution

It provides a complete classification of non-toric $Q$-factorial terminal Fano threefolds with Fano index 13, identifying them as weighted hypersurfaces in a particular weighted projective space.

Findings

Such threefolds are weighted hypersurfaces of degree 12 in $P(3,4,5,6,7)$.

The classification confirms the uniqueness of this geometric structure.

The result advances understanding of Fano threefolds with high Fano index.

Abstract

We show that a non-toric $\mathbb{Q}$-factorial terminal Fano threefold of Picard rank $1$ and Fano index $13$ is a weighted hypersurface of degree $12$ in $\mathbb{P}(3,4,5,6,7)$.