Classification of Okamoto-Painlev\'e Pairs

TL;DR

This paper introduces the concept of Okamoto-Painlevé pairs, provides a new example not previously listed, and offers a complete classification of these pairs, advancing the understanding of their geometric structures.

Contribution

It defines Okamoto-Painlevé pairs, finds a novel example, and achieves a full classification of these geometric objects.

Findings

New example of Okamoto-Painlevé pair identified

Complete classification of Okamoto-Painlevé pairs achieved

Connections to spaces of initial values of Painlevé equations established

Abstract

In this paper, we introduce the notion of an Okamoto-Painlev\'e pair (S, Y) which consists of a compact smooth complex surface S and an effective divisor Y on S satisfying certain conditions. Though spaces of initial values of Painlev\'e equations introduced by K. Okamoto give examples of Okamoto-Painleve pairs, we find a new example of Okamoto-Painlev\'e pairs not listed in \cite{Oka}. We will give the complete classification of Okamoto-Painlev\'e pairs.