Discrete geometry of Cox rings of blow-ups of ℙ³

We prove quadratic generation for the ideal of the Cox ring of the blow-up of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper P cubed"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">P</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb {P}^3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> at <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="7"> <mml:semantics> <mml:mn>7</mml:mn> <mml:annotation encoding="application/x-tex">7</mml:annotation> </mml:semantics> </mml:math> </inline-formula> 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 <italic>Mukai edge graph</italic> .