The cylinder over the Russell cubic threefold is flexible

Abstract

The Russell cubic threefold is the hypersurface $\mathcal{X}\subset {\mathbb{A}}_{\mathbb{C}}^4$ defined by $x^{2}y+z^2+x+t^3=0$. We prove that its cylinder $\mathcal{X}\times {\mathbb{A}}^1$ is flexible. More precisely, after replacing its standard embedding in ${\mathbb{A}}^5$ by an equivalent one, we construct four explicit algebraic ${\mathbb{G}}_a$-actions on the ambient affine space whose restrictions generate a group acting transitively on the cylinder.