Positivity of line bundles on general blow ups of $\mathbb{P}^2$

TL;DR

This paper investigates the positivity properties of line bundles on the blow-up of the projective plane at general points, providing criteria for their ampleness, very ampleness, and global generation.

Contribution

It offers new conditions for the positivity of line bundles on blow-ups of ^2 at general points, advancing understanding of their geometric properties.

Findings

Criteria for ampleness of line bundles on blow-ups

Conditions for very ampleness and global generation

Applications to the geometry of blown-up surfaces

Abstract

Let $X$ be the blow up of $\mathbb{P}^2$ at $r$ general points $p_1,\ldots,p_r \in \mathbb{P}^2$. We study line bundles on $X$ given by plane curves of degree $d$ passing through $p_i$ with multiplicity $m_i$. We establish conditions for ampleness, very ampleness and global generation of such line bundles.