Positivity of line bundles on general blow-ups of abelian surfaces

TL;DR

This paper establishes criteria for the positivity and k-very ampleness of line bundles on blow-ups of abelian surfaces and extends results to surfaces with Picard number one and trivial canonical divisor.

Contribution

It provides new criteria for k-very ampleness of line bundles on blow-ups of abelian surfaces, including cases with Picard number one and trivial canonical divisor.

Findings

Criteria for k-very ampleness of line bundles on blow-ups

Extension of results to surfaces with Picard number one

Analysis of positivity conditions for line bundles

Abstract

Let $(S,L_{S})$ be a polarized abelian surface, and let $M = c \cdot \pi^*L_S - \alpha \cdot \sum_{i=1}^r E_i$ be a line bundle on ${\rm Bl}_{r}(S)$, where $\pi:{\rm Bl}_{r}(S) \rightarrow S$ is the blow-up of $S$ at $r$ general points with exceptional divisors $E_{1},\dots,E_{r}$. In this paper, we provide a criterion for $k$-very ampleness of $M$. Also, we deal with the case when $S$ is an arbitrary surface of Picard number one with a numerically trivial canonical divisor.