Cylinders in Weak Del Pezzo Fibrations

Cylinders in canonical del Pezzo fibrations

“…Then we associate to S "Degree", which is the degree of S, "Singularities", which means singularities on the surface given by the blow-down of all (−2)-curves on S k , and "# Lines", which is the number of all (−1)curves on S k . The triplet (Degree, Singularities, # Lines) is called the type of S (see [28,Section 2.3]). The type of weak del Pezzo surfaces is classified, see for instance [3,7,30], [9,, and [28,Table 3] for the list in the classification.…”

Notes on cylinders in smooth projective surfaces

“…In section 4, we prove Theorem 1.5. The idea of this proof is mainly based on the author's previous work [15]. In section 5, we prove Theorem 1.7.…”

“…Moreover, the author recently gives the existing condition of cylinders in smooth minimal del Pezzo surfaces of Picard rank two. These results are summarized as the following theorem: [15,Theorem 1.7]). Let V be a smooth minimal del Pezzo surface defined over a perfect field k. Then the following assertions hold:…”

“…[6] and [15] only deal with the case of characteristic zero, but note that Theorem 1.3 holds for an arbitrary characteristic. Indeed, their arguments work verbatim even if char(k) > 0.…”

“…It is enough to treat only the case of minimal geometrically rational because the other cases are not difficult (see Remark 1.6). [15] also gives the existence condition of cylinders in minimal weak del Pezzo surfaces with its anti-canonical divisor not ample. By generalizing this work to minimal geometrically rational projective surfaces over perfect fields, we have the first main result as follows:…”