Okawa and Ueda (2025) introduced marked Artin–Schelter surfaces of del Pezzo types — noncommutative surfaces presented as qgr of Artin–Schelter-regular Z\mathbb{Z}-algebras associated with helices on del Pezzo surfaces — and proved that the moduli stack MQ\mathcal{M}_Q of such surfaces of a fixed type is birational to the moduli stack Mell\mathcal{M}_{\mathrm{ell}} of tuples (E,L0,L1,M1,…,Mn)(E, L_0, L_1, M_1,\dots,M_n) consisting of a genus-one curve, two line bundles of degree 3, and line bundles of degree 1. The proof is indirect, via mutations of exceptional and spherical collections, and the birational correspondence is established only on Zariski-open loci of sufficiently general tuples and potentials. The paper states as an expectation, left open, that the birational inverse MQ⇢Mell\mathcal{M}_Q\dashrightarrow\mathcal{M}_{\mathrm{ell}} is concretely given through the moduli scheme of point representations of the directed subquiver Q→⊂QQ^\rightarrow\subset Q, as was proved for cubic surfaces by Abdelgadir–Okawa–Ueda. The task is to prove this point-representation realization for all del Pezzo types, thereby turning the generic, abstract birational equivalence into an explicit and global moduli correspondence.

Background

A del Pezzo surface is a smooth projective surface whose anti-canonical divisor is ample; over an algebraically closed field it is either P1×P1\mathbb{P}^1\times\mathbb{P}^1 or the plane blown up at nn points in general position, with degree d=KX2=9−nd=K_X^2=9-n. A marking is a choice of isomorphism between a fixed lattice Λ\Lambda and the Picard lattice of XX sending a distinguished element to the canonical class, so that moduli of marked del Pezzo surfaces can be viewed as configuration spaces of points on P2\mathbb{P}^2. Noncommutative projective planes are studied through Artin–Schelter (AS) regular algebras: graded algebras of global dimension 3 whose homological behavior mimics the polynomial ring. Writing qgr S=gr S/tor S\mathrm{qgr}\,S=\mathrm{gr}\,S/\mathrm{tor}\,S for the quotient of graded modules by finite-dimensional ones, a noncommutative P2\mathbb{P}^2 is such a category qgr S\mathrm{qgr}\,S. The classification theorem (Artin–Tate–Van den Bergh; Bondal–Polishchuk) states that a noncommutative P2\mathbb{P}^2 is determined by a triple (E,L0,L1)(E, L_0, L_1): a cubic curve EE (the elliptic curve of point modules) and two line bundles of degree 3 on it, and this is the prototype of the geometry–noncommutative-algebra correspondence studied here. Okawa and Ueda formulate the higher-genus-one picture using Z\mathbb{Z}-algebras, i.e. small kk-linear categories with object set Z\mathbb{Z}: an AS-regular Z\mathbb{Z}-algebra of dimension 3 is a connected positively graded Z\mathbb{Z}-algebra with polynomial growth, finite global dimension, and the Artin–Schelter Gorenstein duality condition, and one again forms the quotient category qgr A\mathrm{qgr}\,A. On geometric surfaces one uses exceptional collections: a sequence of objects with Hom(E,E)=k\mathrm{Hom}(E,E)=k and no morphisms backwards, generating the bounded derived category (the homological bookkeeping category built from sheaves on the surface); a helix is a bi-infinite periodic extension of such a collection (period ℓ=12−d\ell=12-d on a del Pezzo surface of degree dd), and it is very strong if Hom⁡k(Ei,Ej)=0\operatorname{Hom}^k(E_i,E_j)=0 for all −∞<i<j<∞-\infty<i<j<\infty and all k≠0k\neq 0 (Okawa–Ueda, Definition 3.1). The rolled-up helix algebra is the total morphism algebra of one period, graded by the period index; it is a Calabi–Yau algebra of dimension 3 in the sense of Ginzburg — an algebra carrying noncommutative Poincaré duality, the analog of a trivial canonical bundle — and every such graded algebra with a finite presentation comes from a quiver with potential, i.e. a quiver together with a linear combination of cyclic paths whose formal cyclic derivatives generate the relations (Bocklandt; Van den Bergh; related tilting Calabi–Yau algebras from del Pezzo helices were studied by Bridgeland–Stern). For each del Pezzo deformation type this produces a rolled-up quiver QQ; the moduli stack PQ\mathcal{P}_Q of potentials of QQ is the quotient stack of the affine space parametrizing coefficients of primitive cyclic paths by the natural gauge group, and the moduli stack MQ\mathcal{M}_Q of marked Artin–Schelter surfaces of type QQ is the open substack where the associated Z\mathbb{Z}-algebra is AS-regular. The geometric side is the stack Mell\mathcal{M}_{\mathrm{ell}} of tuples (E,L0,L1,M1,…,Mn)(E, L_0, L_1, M_1,\dots,M_n) with EE a smooth genus-one curve, L0,L1L_0,L_1 line bundles of degree 3, and M1,…,MnM_1,\dots,M_n line bundles of degree 1; for P2\mathbb{P}^2 (n=0n=0) this reduces to the classical classifying triple. Okawa and Ueda construct rational maps AQ ⁣:Mell⇢MQ\mathcal{A}_Q\colon\mathcal{M}_{\mathrm{ell}}\dashrightarrow\mathcal{M}_Q (from the spherical collection obtained by spherical twists on EE; a spherical object is one whose self-Ext groups reproduce the cohomology algebra of a circle — on a genus-one curve, degree-zero line bundles and skyscraper sheaves are the basic examples — and a spherical twist is the autoequivalence of the derived category that such an object induces) and BQ ⁣:MQ⇢Mell\mathcal{B}_Q\colon\mathcal{M}_Q\dashrightarrow\mathcal{M}_{\mathrm{ell}} (from mutations of the projective helix of the AS-regular Z\mathbb{Z}-algebra, identifying a noncommutative-P2\mathbb{P}^2 block via its classifying triple and point objects) and prove in their Theorem 6.4 that these are mutually inverse on suitable loci — but only for sufficiently general tuples and potentials, with genericity used through upper semicontinuity of cohomology dimensions. The models for an explicit inverse are the known special cases: noncommutative quadrics (Van den Bergh), the one-point blow-up / Hirzebruch case (Mori–Okawa–Ueda), and above all the cubic surfaces case (Abdelgadir–Okawa–Ueda), where Theorem 1.1 exhibits the inverse through the projective moduli scheme Nθ\mathcal{N}_\theta of θ\theta-stable point representations of the quiver with relations (representations of dimension vector (1,…,1)(1,\dots,1), i.e. one-dimensional vector spaces at each vertex, carrying universal line bundles). A point module is the Z\mathbb{Z}-algebra analog of the graded module cutting out a point of the surface; point representations of the directed subquiver Q→Q^\rightarrow (the subquiver of arrows going from lower to higher vertex labels) are precisely the modules expected to parametrize the elliptic curve and the points of the blow-up. Two structural facts frame the problem: a del Pezzo surface admits a very strong helix of line bundles only in degree d≥3d\ge 3 (Hille–Perling), so for d≤2d\le 2 the helix necessarily involves vector bundles and the quiver analysis is harder; and the surfaces obtained are noncommutative blow-ups of noncommutative planes in the sense of Van den Bergh's memoir, whereas Etingof–Ginzburg developed a parallel approach to noncommutative del Pezzo surfaces through Calabi–Yau algebras and local deformation theory, which is local in both base and fiber where the moduli approach is global — the relation between the two approaches is itself raised as an open problem in both sources (Etingof–Ginzburg, Section 1.3, state the expectation that the affine part of a six-point blow-up of a quantum plane yields their E6E_6 deformation).

Problem Statement

Work over an algebraically closed field. Let QQ be the rolled-up helix quiver of a very strong helix on a del Pezzo surface of degree dd (period ℓ=12−d\ell=12-d), MQ\mathcal{M}_Q the moduli stack of marked Artin–Schelter surfaces of type QQ, i.e. the open substack of the moduli stack of potentials of QQ parametrizing potentials whose associated Z\mathbb{Z}-algebras are AS-regular, and Mell\mathcal{M}_{\mathrm{ell}} the moduli stack of tuples (E,L0,L1,M1,…,Mn)(E, L_0, L_1, M_1,\dots,M_n) with EE a smooth projective curve of genus 1, L0,L1L_0, L_1 line bundles of degree 3 on EE, and M1,…,MnM_1,\dots,M_n line bundles of degree 1 on EE (n=9−dn=9-d). Okawa and Ueda constructed rational maps AQ ⁣:Mell⇢MQ\mathcal{A}_Q\colon\mathcal{M}_{\mathrm{ell}}\dashrightarrow\mathcal{M}_Q and BQ ⁣:MQ⇢Mell\mathcal{B}_Q\colon\mathcal{M}_Q\dashrightarrow\mathcal{M}_{\mathrm{ell}} that are mutually inverse on suitable Zariski-open loci of sufficiently general tuples and potentials. The problem, whose single objective is to make this generic birational equivalence explicit and global, is the point-representation realization of the inverse: for every del Pezzo type QQ, construct concretely a moduli scheme of point representations of the directed subquiver Q→⊂QQ^{\rightarrow}\subset Q, with its universal family, and prove that the rational map it induces from MQ\mathcal{M}_Q to Mell\mathcal{M}_{\mathrm{ell}} is a birational inverse of AQ\mathcal{A}_Q, recovering the tuple (E,L0,L1,M1,…,Mn)(E,L_0,L_1,M_1,\dots,M_n) functorially from the universal family. The genericity hypotheses of Okawa–Ueda's Theorem 6.4 (very-strongness of the spherical helix and the required cohomology vanishings, used through upper semicontinuity of cohomology dimensions) are background for this construction; the source raises no question about the complementary non-generic loci, and their behavior is not part of the problem asked here.

References

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