Singularities of Cox Rings of Fano Varieties

Abstract

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

15 pages, 2 tikz figures. Added log Fano and log Calabi-Yau cases, and included an example of a non Cohen-Macaulay Cox ring due to Yoshinori Gongyo

References in corpus (2)

• The Cox Ring of a Del Pezzo Surface Has Rational Singularities
• Characterization of varieties of Fano type via singularities of Cox rings

Cited by in corpus (3)

• Asymptotically log Fano varieties
• Characterization of varieties of Fano type via singularities of Cox rings
• Fano threefolds with 2-torus action - a picture book