Singularities of Cox Rings of Fano Varieties

TL;DR

This paper proves that the Cox ring of a smooth complete Fano variety over complex numbers is Gorenstein with canonical singularities, revealing important structural properties of these algebraic objects.

Contribution

It establishes that the Cox ring of such Fano varieties has Gorenstein and canonical singularities, a new insight into their algebraic structure.

Findings

Cox ring is Gorenstein

Cox ring has canonical singularities

Applicable to smooth complete Fano varieties over complex numbers

Abstract

Let $X$ be a smooth complete Fano variety over $\mathbb{C}$. We show that the Cox ring $\bigoplus_{L\in\text{Pic}(X)}H^0(X,\mathcal{O}_X(L))$ is Gorenstein with canonical singularities.