Klt del Pezzo surfaces which are not Globally F-split

Abstract

Abstract. We construct a klt del Pezzo surface which is not globally F -split, over any algebraically closed field of positive characteristic.

New directions in the Minimal Model Program

“…On the other hand, klt singularities are not necessarily strongly F-regular (e.g. see [18,19]) and, therefore, we cannot consider the category of strongly F-regular pairs as a good candidate of the vertices for our directed graph. Indeed, it is easy to construct examples of strongly F-regular pairs such that their minimal model is not strongly F-regular.…”

On log del Pezzo surfaces in large characteristic

“…For the definitions of -singularities, we refer to [SS10, Definition 3.1] and [CTW17, Definition 1.6].…”

“…Unfortunately, if is numerically trivial, then does not need to be globally -split [CTW17, Theorem 1.1]. Thus, we need to consider two different cases, depending on whether is plt or not.…”

“…Thus, after possibly replacing by a larger number depending only on , by [CGS16, Theorem 1.1], we may assume that is strongly -regular. By [CTW17, Proposition 3.3], is -pure, hence it is sharply -pure by part (a). Thus, part (b) holds.…”

“…Thus, it is natural to ask whether this result can be generalised to higher-dimensional varieties. Unfortunately, in [CTW17] we give a negative answer to this question. Indeed, we show that over an arbitrary algebraically closed field of characteristic , there exists a projective klt surface over such that is ample, but is not globally -regular.…”