Non-log liftable log del Pezzo surfaces of rank one in characteristic five
Masaru Nagaoka
TL;DR
The paper classifies rank-one log del Pezzo surfaces over an algebraically closed field of characteristic five that fail log liftability or have singularities infeasible in characteristic zero, building on Lacini’s and Kawakami–Kamiya–Katz–KN frameworks. It introduces the pathologies (ND),(NB),(NK),(NL), proves the chain $(NK) ightarrow(ND) ightarrow(NL)$ for rank-one surfaces in $p=5$, and provides a complete Dynkin-type criterion for NL, including exceptional families and a $P^1_k$-parametrization in a specific Dynkin-type family. It further constructs three new non-log-liftable examples without tigers in char $5$, refines hunt-step methods via almost log canonical singularities to analyze tiger extractions, and classifies the possible outcomes into finite LDP classes. Finally, it establishes Kawamata–Viehweg vanishing for ample $Z$-Weil divisors when the corresponding singularities are feasible in characteristic zero, linking liftability obstructions to vanishing properties and providing a comprehensive view of liftability phenomena for rank-one log del Pezzo surfaces in characteristic five.
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.