Affine cones over Fano threefolds and additive group actions

Abstract

In this paper we address the following questions for smooth Fano threefolds of Picard number 1: \begin{itemize} \item \textit{When does such a threefold $X$ possess an open cylinder $U \simeq Z\times\mathbb{A}^{1}$, where $Z$ is a surface?} \item \textit{When does an affine cone over $X$ admit an effective action of the additive group of the base field?} \end{itemize} A geometric criterion from [26] (see also [27]) says that the two questions above are equivalent. In [26] we found some interesting families of Fano threefolds carrying a cylinder. Here we provide new such examples.