Takashi Kishimoto


Cylinders in del Pezzo fibrations


For a special kind of affine algebraic variety, the existence of a unipotent group action on it can be interpreted by means of projective geometry. More precisely, letting $(V,H)$ be a polarized variety, the affine cone associated to $(V,H)$ admits an effective $G_a$-action if and only if $V$ admits an $H$-polar cylinder. This fact motivates our interest to find cylinders in various projective varieties. From the viewpoint of minimal model theory, it is more essential to observe cylinders in Mori fiber spaces. In the previous work with Prokhorov and Zaidenberg, we constructed several non-trivial examples of cylinders found in smooth Fano threefold of Picard number one. In the present talk, we shall investigate cylinders in del Pezzo fibrations, which are the other representative of threefolds Mori fiber spaces. The consideration is different according to the degree of del Pezzo fibrations, which ranges over the values from one to nine except for seven, as well known. For the case of degree greater than 4, we can always find "vertical" cylinders, whereas for the case of degree smaller than or equal to 3, there never exists vertical one. Instead, we construct some non-vertical cylinders found in special del Pezzo fibrations of degree 1,2, and 3 respectively.