Del Pezzo surfaces and Mori fiber spaces in positive characteristic

Abstract

We settle a question that originates from results and remarks by Kollár on extremal ray in the minimal model program: In positive characteristics, there are no Mori fibrations on threefolds with only terminal singularities whose generic fibers are geometrically non-normal surfaces. To show this we establish some general structure results for del Pezzo surfaces over imperfect ground fields. This relies on Reid's classification of non-normal del Pezzo surfaces over algebraically closed fields, combined with a detailed analysis of geometrical nonreducedness over imperfect fields of p-degree one.

On Del Pezzo Fibrations in Positive Characteristic

“…By Lemma 6.1, we have p-deg(k) 1. Therefore, X is geometrically normal by [11,Theorem 14.1]. Thus, we conclude by Propositions 6.4, 6.5, and 6.6.…”

“…In this case, X is a regular del Pezzo surface with ρ(X ) = 1. Since the p-degree of a C 1 -field is at most one (Lemma 6.1), it follows from [11,Theorem 14.1] that X is geometrically normal. Then Theorem 1.5 implies that the base change X × k k is a canonical del Pezzo surface, i.e.…”

“…Furthermore, Fanelli and Schröer show that a regular del Pezzo surface X is geometrically normal in every characteristic p if [k : k p ] p and ρ(X ) = 1 (cf. [11,Theorem 14.1]).…”

The maximal unipotent finite quotient, unusual torsion in Fano threefolds, and exceptional Enriques surfaces

“…In [Sch07], related problems where considered for non-normal del Pezzo surface. Del Pezzo surfaces over F with pdeg(F) = 1 were studied in [FS20]. Here the p-degree is defined as pdeg(F) = dim F (Ω 1 F/F p ).…”

“…In turn, the above examples of non-normal Fano threefolds Y may admit twisted forms Y whose local rings are normal Q-factorial klt singularities, and thus could occur as generic fibers in Mori fiber spaces. See [FS20] for our analysis of non-normal del Pezzo surfaces having twisted forms whose local rings are regular.…”