The $d$-very ampleness of adjoint line bundles on quasi-elliptic surfaces

TL;DR

This paper establishes a numerical criterion for the $d$-very ampleness of adjoint line bundles on quasi-elliptic surfaces and provides a new proof of an optimal vanishing theorem for these surfaces.

Contribution

It introduces a Reider-type numerical criterion for $d$-very ampleness and offers a new, optimal proof of the vanishing theorem on quasi-elliptic surfaces.

Findings

Established a Reider-type criterion for $d$-very ampleness.

Provided a new proof of the vanishing theorem.

Showed the vanishing theorem is optimal.

Abstract

In this paper, we give a numerical criterion of Reider-type for the $d$-very ampleness of the adjoint line bundles on quasi-elliptic surfaces, and meanwhile we give a new proof of the vanishing theorem on quasi-elliptic surfaces emailed from Langer and show that it is the optimal version.