Flexibility of affine cones over a smooth complete intersection of two quadrics
Kirill Shakhmatov, Hoang Le Truong
TL;DR
The paper proves that affine cones over smooth complete intersections of two quadrics are flexible, extending the reach of unipotent automorphism actions in algebraic geometry. It develops a robust cylinder-based framework: first establishing flexibility for complements of quadrics, then transferring the property to affine cones via a birational model and transversal cylinder coverings. The key contribution is a geometric criterion linking transversal cylinder coverings to flexibility and applying it to two-quadratic complete intersections. This work broadens the class of known flexible affine varieties and strengthens the connection between cylinder structures and the automorphism groups of Fano-type spaces.
Abstract
We prove flexibility of two families of affine varieties: the complement in $\mathbb{P}^n$ of a projective quadric of rank at least three and affine cones over a smooth complete intersection of two quadrics in $\mathbb{P}^{n + 2}$, $n \ge 3$.