AFFINE CONES OVER FANO THREEFOLDS AND ADDITIVE GROUP ACTIONS

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