Smooth Rational Curves on Singular Rational Surfaces

TL;DR

This paper classifies certain complex surfaces with quotient singularities lacking smooth rational curves, and shows that specific log del Pezzo surfaces must contain at least one smooth rational curve.

Contribution

It provides a classification of singular rational surfaces without smooth rational curves under certain conditions and establishes the existence of such curves in specific log del Pezzo surfaces.

Findings

Classified all complex surfaces with quotient singularities without smooth rational curves.

Proved that certain log del Pezzo surfaces necessarily contain smooth rational curves.

Established conditions under which smooth rational curves must exist on singular surfaces.

Abstract

We classify all complex surfaces with quotient singularities that do not contain any smooth rational curves, under the assumption that the canonical divisor of the surface is not pseudo-effective. As a corollary we show that if $X$ is a log del Pezzo surface such that for every closed point $p\in X$, there is a smooth curve (locally analytically) passing through $p$, then $X$ contains at least one smooth rational curve.