Non-log liftable log del Pezzo surfaces of rank one in characteristic five

TL;DR

This paper classifies certain rank one log del Pezzo surfaces in characteristic five, focusing on those not liftable to characteristic zero or with incompatible singularities, and confirms vanishing theorems under specific conditions.

Contribution

It extends the classification of rank one log del Pezzo surfaces in characteristic five and analyzes their liftability and singularity properties.

Findings

Identifies non-log liftable surfaces in characteristic five.

Shows vanishing theorem holds for feasible singularities.

Provides isomorphism class classification for these surfaces.

Abstract

Building upon the classification by Lacini [arXiv:2005.14544], we determine the isomorphism classes of log del Pezzo surfaces of rank one over an algebraically closed field of characteristic five either which are not log liftable over the ring of Witt vectors or whose singularities are not feasible in characteristic zero. We also show that the Kawamata-Viehweg vanishing theorem for ample $\mathbb{Z}$-Weil divisors holds for log del Pezzo surfaces of rank one in characteristic five if those singularities are feasible in characteristic zero.