On Orevkov's rational cuspidal plane curves

TL;DR

This paper classifies certain rational cuspidal plane curves with one cusp and maximal self-intersection, showing they match those constructed by Orevkov, thus advancing understanding of their geometric properties.

Contribution

It provides a classification of rational cuspidal plane curves with specific properties, confirming they are exactly the curves constructed by Orevkov.

Findings

Curves with the specified properties are classified and shown to coincide with Orevkov's constructions.

The strict transforms of these curves have maximal self-intersection numbers.

The classification enhances understanding of the geometry of rational cuspidal plane curves.

Abstract

In this note, we consider rational cuspidal plane curves having exactly one cusp whose complements have logarithmic Kodaira dimension two. We classify such curves with the property that the strict transforms of them via the minimal embedded resolution of the cusp have the maximal self-intersection number. We show that the curves given by the classification coincide with those constructed by Orevkov.