A review of Lacini's classification of rank one log del Pezzo surfaces in characteristic different from two and three

TL;DR

This paper reviews Lacini's classification of rank one log del Pezzo surfaces in characteristics other than two and three, highlighting potential omissions and extending previous results to all characteristics.

Contribution

It critically examines Lacini's classification, identifies a possible missing surface, and generalizes results to arbitrary characteristic.

Findings

Identification of a potentially omitted surface in Lacini's list.

Extension of Lacini's results to all characteristics.

Clarification of techniques used in classification.

Abstract

This paper reviews Lacini's classification [Lac24] of log del Pezzo surfaces of rank one in characteristics different from two and three, with a focus on where and how Lacini enhanced the techniques of Keel and McKernan [KM99]. We point out that there is at most one log del Pezzo surface that may have been erroneously omitted from the list in [Lac24, \S 6.1], namely the surface in Example 3.13. We also extend the results of [Lac24 \S 4.2] to arbitrary characteristic.