# Bounded negativity and Harbourne constants on ruled surfaces

## TL;DR

This paper investigates the bounded negativity conjecture by establishing lower bounds for Harbourne constants on ruled surfaces, contributing to understanding negative self-intersections of curves on algebraic surfaces.

## Contribution

It provides new lower bounds for Harbourne constants specifically on geometrically ruled surfaces, advancing the study of negative self-intersection curves.

## Key findings

- Lower bounds for Harbourne constants on ruled surfaces are established.
- Results support the bounded negativity conjecture in the context of ruled surfaces.
- The work connects curve arrangements with negativity bounds on algebraic surfaces.

## Abstract

Let $X$ be a smooth projective surface and let $\mathcal{C}$ be an arrangement of curves on $X$. The Harbourne constant of $\mathcal{C}$ was defined as a way to investigate the occurrence of curves of negative self-intersection on blow ups of $X$. This is related to the bounded negativity conjecture which predicts that the self-intersection number of all reduced curves on a surface is bounded below by a constant. We consider a geometrically ruled surface $X$ over a smooth curve and give lower bounds for the Harbourne constants of transversal arrangements of curves on $X$.