Log canonical thresholds of Del Pezzo Surfaces in characteristic p

TL;DR

This paper extends the computation of global log canonical thresholds from complex del Pezzo surfaces to those over algebraically closed fields of positive characteristic, introducing new algebraic techniques and establishing K-semistability in low degrees.

Contribution

It provides algebraic proofs for log canonical thresholds of del Pezzo surfaces in characteristic p, replacing complex-analytic methods with a classification-based approach.

Findings

Computed thresholds for del Pezzo surfaces in characteristic p

Established K-semistability for low-degree surfaces in positive characteristic

Introduced a new classification technique for low-degree curves

Abstract

The global log canonical threshold of each non-singular complex del Pezzo surface was computed by Cheltsov. The proof used Koll\'ar-Shokurov's connectedness principle and other results relying on vanishing theorems of Kodaira type, not known to be true in finite characteristic. We compute the global log canonical threshold of non-singular del Pezzo surfaces over an algebraically closed field. We give algebraic proofs of results previously known only in characteristic $0$. Instead of using of the connectedness principle we introduce a new technique based on a classification of curves of low degree. As an application we conclude that non-singular del Pezzo surfaces in finite characteristic of degree lower or equal than $4$ are K-semistable.