The cylinder over the Russell cubic threefold is flexible

Abstract

The Russell cubic threefold is the hypersurface $\mathcal X\subset\mathbb A^4_{\mathbb C}$ defined by $x^2y+z^2+x+t^3=0$. 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.