Negative curves on special rational surfaces

TL;DR

This paper investigates negative curves on certain rational surfaces formed by blowing up special point configurations, demonstrating the Bounded Negativity Conjecture for these cases and proposing future research directions.

Contribution

It provides new insights into negative curves on rational surfaces with specific point configurations and confirms the Bounded Negativity Conjecture in these contexts.

Findings

Bounded Negativity Conjecture holds for the studied surfaces

Negative curves are characterized for special point configurations

Analysis includes points on a cubic, 3-torsion points, and Fermat points

Abstract

We study negative curves on surfaces obtained by blowing up special configurations of points in the complex projective palne. Our main results concern the following configurations: very general points on a cubic, 3-torsion points on an elliptic curve and nine Fermat points. As a consequence of our analysis, we also show that the Bounded Negativity Conjecture holds for the surfaces we consider. The note contains also some problems for future attention.