Spherical actions on locally factorial Fano varieties of dimension $\leq 4$ and rank $\leq 2$

TL;DR

This paper classifies all faithful spherical actions of rank two or less on locally factorial Fano varieties of dimension four or less, providing a comprehensive list and detailed combinatorial data for these geometric objects.

Contribution

It provides the complete classification of spherical actions and explicit combinatorial data for locally factorial Fano varieties of small dimension and rank, advancing understanding of their symmetry and structure.

Findings

List of 337 faithful spherical actions of rank ≤ 2

Explicit classification of spherical homogeneous spaces of dimension ≤ 4

Determination of combinatorial data for locally factorial G/H-reflexive polytopes

Abstract

We obtain the exhaustive list of 337 faithful spherical actions of rank two or less on locally factorial Fano manifolds of dimension four or less. As a preliminary step, we determine the explicit list of spherical homogeneous spaces of dimension four or less, together with their combinatorial data. Then we classify the possible locally factorial $G/H$-reflexive polytopes for each such spherical homogeneous space $G/H$. From the combinatorial data gathered in this article, one can easily read off the Picard rank (even the Picard group), Fano index, anticanonical volume of the underlying locally factorial Fano variety, etc.