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

TL;DR

This paper investigates the positivity properties of line bundles on blow-ups of the projective plane at points on a given curve, providing criteria for ampleness and k-very ampleness.

Contribution

It establishes new conditions for the ampleness and k-very ampleness of line bundles on special blow-ups of bb2 at points on a curve.

Findings

Derived criteria for ampleness of line bundles.

Established conditions for k-very ampleness.

Analyzed the impact of blow-up points on line bundle positivity.

Abstract

Let $C \subset \mathbb{P}^2$ be an irreducible and reduced curve of degree $e > 0$. Let $X$ be the blow up of $\mathbb{P}^2$ at $r$ distinct smooth points $p_1,\ldots,p_r \in C$. We study line bundles on $X$ and establish conditions for ampleness and $k$-very ampleness.