Cox ring of the blow up of $7$ points in $\mathbb{P}^3$

This repository contains the auxiliary code to the paper Discrete Geometry of Cox Rings of blow-ups of $\mathbb{P}^3$ by Mara Belotti and Marta Panizzut.

About the paper

The Arxiv identifier is 2208.05258.
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.

About the code

The helper_functions.jl and cox_generators.jl contains auxiliary code for the main functions in is_khovanskii.jl. This is a quick example on how to use the code in a julia terminal:

julia> KS, (t,z...)=PolynomialRing(QQ, ["t",["z[$i]" for i in 1:7]...]);
julia> P=[t^59 t^44 t^79 t^20 t^12 t^81 t^36; t^8 t^72 t^49 t^39 t^58 t^23 t^64; t^44 t^58 t^12 t^52 t^57 t^49 t^51; t^25 t^23 t^60 t^72 t^45 t^51 t^6];
julia> bool,_,_=generalposition_monomeric_check(P);
julia> bool
true

The necessary code for the proof of Theorem 12 is contained in the tutorial.jl jupyter notebook. The file Lemma22.ipynb contains the code necessary for the proof of Corollary 22.

References

Mara Belotti, Marta Panizzut. Discrete geometry of Cox rings of blow-ups of $\mathbb{P}^3$, arXiv:2208.05258.