Rationality of Seshadri constants on blow-ups of ruled surfaces

TL;DR

This paper investigates Seshadri constants on blow-ups of ruled surfaces, proposing a conjecture for classifying negative curves and demonstrating the existence of irrational Seshadri constants under this conjecture.

Contribution

It introduces a conjecture for negative curve classification on blown-up ruled surfaces and shows irrational Seshadri constants can occur assuming the conjecture.

Findings

Proposes a conjecture for negative self-intersection curves.

Shows existence of irrational Seshadri constants.

Extends previous work on Hirzebruch surfaces.

Abstract

In this note, we continue the study of Seshadri constants on blow-ups of Hirzebruch surfaces initiated in arXiv:2312.14555. Now we consider blow-ups of ruled surfaces more generally. We propose a conjecture for classifying all the negative self-intersection curves on the blow-up of a ruled surface at very general points, analogous to the $(-1)$-curves conjecture in $\mathbb{P}^2$. Assuming this conjecture is true, we exhibit an ample line bundle with an irrational Seshadri constant at a very general point on such a surface.