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

Abstract

In this paper, we study pathologies of Du Val del Pezzo surfaces defined over an algebraically closed field of positive characteristic by relating them to their non-liftability to the ring of Witt vectors. More precisely, we investigate 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, Picard rank, and anti-canonical degree, (NK): there exists an ample $\mathbb{Z}$-divisor which violates the Kodaira vanishing theorem for $\mathbb{Z}$-divisors, 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. As a result, for each of these conditions, we determine all the Du Val del Pezzo surfaces which satisfy the given one.

43 pages, v3: the final version. According to the referee's suggestion, we made the paper self-contained. The title was slightly changed. To appear in the Mathematische Zeitschrift