Seshadri constants and negative curves on blowups of ruled surfaces

TL;DR

This paper computes Seshadri constants for ample line bundles on blowups of Hirzebruch and certain ruled surfaces, and investigates bounded negativity properties of these blowups.

Contribution

It provides explicit calculations of Seshadri constants on blowups of Hirzebruch and ruled surfaces, extending understanding of their geometric properties.

Findings

Seshadri constants computed for blowups at up to e+3 points

Results on bounded negativity of these blowups

Extension of Seshadri constant calculations to ruled surfaces

Abstract

In this article we compute Seshadri constants of ample line bundles on the blowup of Hirzebruch surface $\mathbb{F}_e$ at $r\leqslant e+3$ very general points. Similarly, we compute Seshadri constants on the blowups of certain decomposable ruled surfaces over smooth curves of non-zero genus. We also prove some results related to bounded negativity of blowups of Hirzebruch surfaces and ruled surfaces.