Discrete geometry of Cox rings of blow-ups of $\mathbb{P}^3$

TL;DR

This paper proves quadratic generation for the Cox ring of the blow-up of projective 3-space at 7 points, using Khovanskii bases and toric degenerations, thus solving a conjecture by Lesieutre and Park.

Contribution

It introduces methods to compute Khovanskii bases for Cox rings of blow-ups of projective space, extending techniques from Del Pezzo surfaces.

Findings

Proved quadratic generation of the Cox ring for the blow-up of P^3 at 7 points.

Developed computational techniques for Khovanskii bases in this context.

Analyzed the associated polytopes and introduced the Mukai edge graph.

Abstract

We prove quadratic generation for the ideal of the Cox ring of the blow-up of $\mathbb{P}^3$ at $7$ points, solving a conjecture of Lesieutre and Park. To do this we compute Khovanskii bases, implementing techniques which proved successful in the case of Del Pezzo surfaces. Such bases give us degenerations to toric varieties whose associated polytopes encode toric degenerations with respect to all projective embeddings. We study the edge-graphs of these polytopes and we introduce the Mukai edge graph.