Cylinders in Fano varieties

Abstract

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Cylinders in canonical del Pezzo fibrations

“…Proof of Proposition 4.9 (3). In this subsection, we shall prove Proposition 4.9 (3). With the notation as in Proposition 4.9(3), notice that E is a (−1)-curve on S k by Proposition 4.9(1).…”

“…Then it is easily to see that the equality (4.7) holds if only and if ((M 1 ) 2 , (M 2 ) 2 ) = (− 5 6 , − 2 3 ) or (− 3 4 , − 3 4 ), here remark n(1) ≥ n(2). This means that (n(1), n(2)) = (5, 2) or (3,3) by Table 3. The other cases are left to the reader because these can be shown by an argument similar to above argument.…”

“…Here, in Case 1, we shall assume (d, r) = (2, 2), (2, 1), (1,3), (1,2) or (1, 1). Furthermore, in Case 2 (resp.…”

“…In (3), this proof is a bit long and is needed technical argument. Hence, we will present this proof in § §4.…”

“…Let X be an algebraic variety over k. Then an open subset U of X is called a cylinder if U is isomorphic to A 1 k × Z for some variety Z over k. Certainly, cylinders are geometrically very simple, however recently they begin to receive a lot of attentions in connection with unipotent group actions on affine cones over polarized varieties (see, e.g., [3,4,12,13,14,15]). Thus, it is important to find a cylinder in projective varieties, but in general it is not easy to decide whether a given projective variety V contains a cylinder.…”