Positivity of line bundles on general blow ups of Hirzebruch surfaces

TL;DR

This paper studies positivity properties of line bundles on general blow ups of Hirzebruch surfaces, providing numerical conditions for ampleness, global generation, very ampleness, and k-very ampleness.

Contribution

It extends the study of positivity properties from blow ups of projective planes to Hirzebruch surfaces, offering new numerical criteria.

Findings

Numerical conditions for ampleness established.

Criteria for global generation derived.

Conditions for very ampleness and k-very ampleness provided.

Abstract

We investigate various positivity properties of line bundles on general blow ups of Hirzebruch surfaces motivated by \cite{Han}, where the author has studied general blow ups of $\mathbb{P}^2$. For each of the properties: ampleness, global generation, very ampleness, and $k$-very ampleness, we provide several sufficient numerical conditions.