March 3, 2026

Discrete geometry of Cox rings of blow-ups of ℙ³

Key Points

Quadratic generation for the ideal of the Cox ring has been established in the blow-up of projective space at seven points.
Key metric involves computing Khovanskii bases, which relate to known results for del Pezzo surfaces.
Analysis covers toric varieties associated with polytopes that delineate projective embeddings—the edge-graphs revealed new structural insights.
Significance for future geometric studies emerges from the introduction of the Mukai edge graph, potentially expanding toric degeneracy frameworks.

Abstract

We prove quadratic generation for the ideal of the Cox ring of the blow-up of P 3 P³ at 7 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.

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Discussion

Authors

Mara Belotti

Marta Panizzut

Journals

Mathematics of Computation

Actions

Institutions

Technische Universität Berlin

Forskning.no (Norway)

References and Citations

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Building similarity graph...

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Discrete geometry of Cox rings of blow-ups of ℙ³

Key Points

Abstract

Citation Network

Connected Papers

Discussion

Authors

Journals

Actions

Institutions

References and Citations

Citation Network

Connected Papers

Discussion

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