Contractible curves on a rational surface

TL;DR

This paper proves that certain divisors on rational surfaces can be contracted when the log-Kodaira dimension is negative infinity, extending to cases without blowing up points and providing insights for non-connected divisors.

Contribution

It establishes conditions under which divisors on rational surfaces are contractible, including cases without point blow-ups and for non-connected divisors, advancing the understanding of surface contractions.

Findings

Contractibility when log-Kodaira dimension is -infinity

Extension to cases without blowing up points

Insights into non-connected divisors using peeling theory

Abstract

In this paper we prove that if S is a smooth, irreducible, projective, rational, complex surface and D an effective, connected, reduced divisor on S, then the pair (S,D) is contractible if the log-Kodaira dimension of the pair is $-\infty$. More generally, we even prove that this contraction is possible without blowing up an assigned cluster of points on S. Using the theory of peeling, we are also able to give some information in the case D is not connected.