Cylinders, flexibility, and quadrics

Given an algebraic variety $X$, by a cylinder in $X$ we mean an open subset isomorphic to the product $\mathbb{A}^1 \times Z$ for some affine variety $Z$. We discuss a connection between cylinders and $\mathbb{G}_a$-actions on affine varieties and provide a criterion of flexibility of an affine variety in terms of cylinders. We prove flexibility of two families of affine varieties: the complement of a quadric in a projective space and an affine cone over a smooth complete intersection of two quadrics in a projective space. The talk is based on a joint work with Hoang Le Truong.