Quadratic Generation for the Canonical-Form Embeddings of X_cf(3, 6) and Y(3,6), and Kozulness for del Pezzo Complements

Jeff Pickhardt

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Quadratic Generation for the Canonical-Form Embeddings of X_cf(3, 6) and Y(3,6), and Kozulness for del Pezzo Complements

Abstract

We resolve Conjecture 5.5 of Hollering–Pavlov–Pratt (Log Canonical Models and Positive Geometries, arXiv:2607.28368) in the affirmative, for both of its assertions. First, the homogeneous vanishing ideal of the canonical-form embedding Xcf (3, 6) ⊆ P125 of the configuration space of six points in P2 is generated by its 6,616-dimensional space of quadrics. Second, the homogeneous vanishing ideal of Naruki’s cross-ratio compactification C⊆P149, the log canonical compactification of the moduli space Y(3, 6) of marked cubic surfaces, is generated by its 9,605-dimensional space of quadrics. Along the way we prove that the log canonical ring of X(3, 6) is generated in degree one, settling a hypothesis left open in the source paper; consequently the canonical-form image coincides with the intrinsic log canonical model. Each proof combines exact intersection theory (giving the Hilbert polynomial of the log canonical polarization), Kawamata–Viehweg vanishing, arithmetic Cohen–Macaulayness and the regularity bound reg I = 5, with exact finite-field certificates — valid in characteristic zero by unit-minor lifting — showing that the quadratic ideal agrees with the vanishing ideal in degrees three, four, and five. Separately, for a smooth complex del Pezzo surface S of degree d ∈{2, 3, 4}with D the reduced sum of all (−1)-curves, we prove that the complete section ring of KS + D is generated in degree one, satisfies property N1, and is Koszul; the complete embeddings have respectively 240, 5,586, and 283,932 minimal quadratic generators. In degree three this gives a conceptual characteristic-zero proof, and a Koszulness strengthening, of the cubic-surface computation of Hollering–Pavlov–Pratt. We do not prove Koszulness of either configuration-space coordinate ring, and we exhibit obstructions showing that the computed quadrics are not a Gr¨obner basis in the tested term orders.

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