An observation on (-1)-curves on rational surfaces

TL;DR

This paper provides an effective iterative method to characterize (-1)-curves on rational surfaces obtained by blowing up the projective plane at general points, based on their intersection properties and genus.

Contribution

It introduces a new iterative characterization of (-1)-curves on rational surfaces, focusing on their intersection numbers and genus, which was not previously formalized.

Findings

Characterization of (-1)-curves via self-intersection and genus.

Iterative method for classifying (-1)-curves.

Application to blowups of the projective plane at general points.

Abstract

We give an effective iterative characterization of the classes of (smooth, rational) (-1)-curves on the blowup of the projective plane at general points. Such classes are characterized as having self-intersection -1, arithmetic genus 0, and intersecting every (-1)-curve of smaller degree nonnegatively.