Prime Fano $4$-folds with semi-free torus actions

TL;DR

This paper classifies smooth prime Fano fourfolds with semi-free complex torus actions, showing they belong to specific known families, and confirms the existence of such actions within these families.

Contribution

It provides a classification of prime Fano fourfolds with semi-free torus actions, identifying the families they belong to and demonstrating the presence of such actions.

Findings

Prime Fano fourfolds with semi-free actions are contained in specific families.

All identified families include members with semi-free $ ext{C}^*$-actions.

The classification links geometric properties to known Fano families.

Abstract

Let $X$ be a smooth complex prime Fano fourfold having a semi-free action of $\mathbb{C}^*$, then $X$ is contained in one of the families $\mathbb{P}^4,Q^4,W_5, X^{m}_{8}$. All of the families contain members that have a semi-free $\mathbb{C}^*$-action.