Nef divisors of surfaces given by pencils at infinity

TL;DR

This paper describes the structure of nef cones and the cone of curves for certain rational surfaces obtained via blow-ups, and establishes finite generation of their Cox rings under specific conditions.

Contribution

It provides explicit generators for nef and curve cones of these surfaces and proves finite generation of the Cox ring in particular cases.

Findings

Generators for nef cone and cone of curves are explicitly described.

Finite generation of Cox ring is proven for surfaces with specific blow-up configurations.

Results apply to surfaces from pencils associated with curves having one place at infinity.

Abstract

We give generators for the nef cone and the cone of curves of rational surfaces obtained by blowing-up the complex projective plane at a set of points $\mathcal{B} \cup \mathcal{D}$, where $\mathcal{B}$ is the set of (proper and infinitely near) base points of a pencil associated with a curve having one place at infinity, and $\mathcal{D}$ is a set of finitely many infinitely near free points on the strict transforms of curves of the pencil. We also prove that, when the pencil is given by an AMS-type curve and $\mathcal{D}$ contains at most two free points on any curve considered, the Cox ring of the obtained surface is finitely generated.