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

TL;DR

This paper investigates the unique pathologies of Du Val del Pezzo surfaces in positive characteristic, linking their singularities and liftability issues to their non-liftability to Witt vectors, and classifies surfaces satisfying specific pathological conditions.

Contribution

It provides a comprehensive classification of Du Val del Pezzo surfaces exhibiting certain pathologies related to singularities and liftability in positive characteristic.

Findings

Characterization of surfaces satisfying condition (NB)

Identification of surfaces with non-liftability to Witt vectors

Classification of surfaces with anti-canonical divisors violating Kodaira vanishing

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.