Fano threefolds as equivariant compactifications of the vector group

TL;DR

This paper classifies all smooth Fano threefolds with Picard number at least two that serve as equivariant compactifications of the three-dimensional vector group, enriching the understanding of algebraic group actions on Fano varieties.

Contribution

It provides a complete classification of such equivariant compactifications among smooth Fano threefolds with Picard number ≥ 2, a previously unexplored area.

Findings

Identified all smooth Fano threefolds with Picard number ≥ 2 that are equivariant compactifications of -dimensional vector group

Established criteria for smoothness and equivariance in these compactifications

Contributed to the classification theory of algebraic group actions on Fano varieties

Abstract

In this article, we determine all equivariant compactifications of the three-dimensional vector group $\mathbf{G}_a^3$ which are smooth Fano threefolds with Picard number greater or equal than two.