Takashi Kishimoto: Cylinders in Mori Fiber Spaces

Cylinders founded in projective varieties provide useful tools to produce effective actions of unipotent groups on affine cones over them. In order to find cylinders in projective varieties, it is in some sense essential to stick to those contained in Mori Fiber Space (MFS) \(\pi:V\rightarrow W\). Cylinders which respect the structure of π are called vertical, whereas those which are not compatible with π are called twisted. Depending on types of cylinders, we are obliged to consider in different fashions to find them in \(V\). For vertical cylinders, the essence lies in behavior of the generic fiber \(V_{\eta}=\pi^{-1}(\eta)\) of \(\pi\), hence generic properties on general fibers of \(\pi\) play an important role, more precisely V contains a vertical cylinder if and only if \(V_{\eta}\) contains a cylinder defined over the function field \(C(\eta)=C(W)\). We will mention some criteria for \(V_{\eta}\) to contain a cylinder for case in which the relative dimension \(r=dim V-dim W\) is less than or equal to four. Meanwhile, for the construction of twisted cylinders, we have to proceed more explicitly: an explicit resolution of certain linear pencils on Fano varities, an explicit description of relative minimal model program (MMP) and an explicit observation of MFS as an outcome of MMP. Especially, we construct a continuous family of compactifications of the affine 4-space \(\mathbb{A}^4_{\mathbb{C}}\) into MFS’s \(\pi:V\rightarrow \mathbb{P}^1_{\mathbb{C}}\) whose general fibers are birationally rigid \(\mathbb{Q}\)-Fano threefolds. The content of lectures is based on the joint work with Adrien Dubouloz (Universit´e de Bourgogne).