Cox rings of rational surfaces and redundant blow-ups

TL;DR

This paper investigates how redundant blow-ups affect the Cox rings of rational surfaces, showing they preserve finite generation and help characterize birational morphisms, leading to new constructions of Mori dream surfaces.

Contribution

It demonstrates that redundant blow-ups preserve Cox ring finite generation and characterizes birational morphisms of Mori dream rational surfaces with specific Iitaka dimensions.

Findings

Redundant blow-ups preserve Cox ring finite generation.

Redundant blow-ups characterize birational morphisms of Mori dream surfaces.

Constructs new Mori dream surfaces with large Picard number.

Abstract

We prove that the redundant blow-up preserves the finite generation of the Cox ring of a rational surface under a suitable assumption, and we study the birational structure of Mori dream rational surfaces via redundant blow-ups. It turns out that the redundant blow-up completely characterizes birational morphisms of Mori dream rational surfaces with anticanonical Iitaka dimension $0$. As an application, we construct new Mori dream rational surfaces with anticanonical Iitaka dimension $0$ and $-\infty$ of arbitrarily large Picard number.