Flexible affine cones and flexible coverings

TL;DR

This paper introduces a new criterion for the flexibility of cones over certain algebraic varieties and demonstrates its application to various classes of varieties, including secant varieties, Fano threefolds, and Cox rings of del Pezzo surfaces.

Contribution

It presents a novel criterion for flexibility of cones and applies it to establish flexibility in several important classes of algebraic varieties.

Findings

Affine cones over secant varieties of Segre--Veronese embeddings are flexible.

Flexibility of affine cones over certain Fano threefolds is proven.

Total coordinate spaces of Cox rings of del Pezzo surfaces are flexible.

Abstract

We provide a new criterion for flexibility of cones over varieties covered by flexible affine varieties. We apply this criterion to prove flexibility of affine cones over secant varieties of Segre--Veronese embeddings and over certain Fano threefolds. We further prove flexibility of total coordinate spaces of Cox rings of del Pezzo surfaces.