Pathologies and liftability of Du Val del Pezzo surfaces in positive characteristic

Abstract

In this paper, we study Du Val del Pezzo surfaces over an algebraically closed field of positive characteristic. Such a surface X may satisfy the condition (NB): all the anti-canonical divisors are singular, (ND): there are no Du Val del Pezzo surfaces over the field of complex numbers with the same Dynkin type and Picard rank, and (NL): the pair (Y, E) does not lift to the ring of Witt vectors, where Y is the minimal resolution and E is its reduced exceptional divisor. On one hand, we show that (ND) ⇒ (NL) ⇒ (NB). On the other hand, for each condition above, we determine all the Du Val del Pezzo surfaces with the given condition. We also give the list of the automorphism groups of Du Val del Pezzo surfaces satisfying one of them. Furthermore, following Ye's method of classification of Du Val del Pezzo surfaces in characteristic zero, we also classify Du Val del Pezzo surfaces of the Picard rank one in characteristic two or three. Finally, we show that if Du Val del Pezzo surfaces are Frobenius split, then they satisfy none of the conditions above.

Rational curves on del Pezzo surfaces in positive characteristic

“…( As shown by Theorem 1.9, the possible weak del Pezzo surfaces S in Theorem 1.12 are the same as the weak del Pezzo surfaces classified by [KN20b,Theorem 1.4]. These examples are quite interesting; they show that over a function field there can be a Zariski dense set of rational points which outpaces the exponential term in the rate predicted by Manin's Conjecture.…”

“…Then we use this result to deduce Theorem 1.12. Finally we found examples of weak del Pezzo surfaces satisfying Theorem 1.12 in [KN20b] and [KN20a] which classified pathological examples of Du Val del Pezzo surfaces. Theorem 1.9 follows from Theorem 1.12 and the analysis of low degree rational curves in Section 3.…”

“…We emphasize that these three types are not mutually exclusive. Our description will be based upon [KN20b] and [KN20a] which classify the weak del Pezzo surfaces in characteristics 2 and 3 such that the anticanonical model has Picard rank 1. [KN20a, Table 1] gives a complete list of such surfaces based on the type of singularities of the anticanonical model; we will refer to this table in our discussion.…”

“…The weak del Pezzo surfaces S for which Theorem 1.1 does not apply -that is, the surfaces S such that every member of | − K S | is singular -are classified by [KN20b]. There are 3 Date: October 5, 2021.…”

An introduction to Geometric Manin's conjecture

“…The classification of these pathological weak del Pezzo surfaces is given in [KN20b] and [KN20a]. The motivation of these papers is the classification of du Val del Pezzo surfaces which do not satisfy Kawamata-Viehweg vanishing or log liftability.…”