Q-Fano threefolds of index $7$

TL;DR

This paper classifies certain $Q$-Fano threefolds with Fano index 7, showing that a large linear system dimension implies the variety is isomorphic to specific weighted projective hypersurfaces.

Contribution

It provides a classification result for $Q$-Fano threefolds of index 7 based on the dimension of their anticanonical linear system.

Findings

If $ ext{dim} |-K_X| extgreater= 15$, then $X$ is isomorphic to one of three specific weighted hypersurfaces.

The classification includes the varieties $P(1^2,2,3)$, $X_6 o P(1,2^2,3,5)$, and $X_6 o P(1,2,3^2,4)$.

Abstract

We show that, for a $\mathbb Q$-Fano threefold $X$ of Fano index 7, the inequality $\dim |-K_X| \ge 15$ implies that $X$ is isomorphic to one of the following varieties $\mathbb P (1^2,2,3)$, $X_6 \subset \mathbb P (1,2^2,3,5)$ or $X_6 \subset \mathbb P (1,2,3^2,4)$.