# 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).