Dolgachev proved that, for any field k, the ring naturally associated to a
generic Laurent polynomial in d variables, $d \geq 4$, is factorial. We prove a
sufficient condition for the ring associated to a very general complex Laurent
polynomial in d=3 variables to be Q-factorial.Comment: 5 page

Antonella Grassi

Batyrev

Dolgachev

Khovanskiĭ

Saito

Ugo Bruzzo

English

arXiv

Dolgachev proves that the ring naturally associated to a generic Laurent polynomial in d variables, d&lt;4, is factorial (for any field k). We prove a sufficient condition for the ring associated to a very general complex Laurent polynomial in d=3 variables to be Q-factorial

Bruzzo, U.

Grassi, A.

Sissa Digital Library

Q-factorial Laurent rings

Crossref

Journal of Pure and Applied Algebra

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Dolgachev proves that the ring naturally associated to a generic Laurent polynomial in d variables, d≥ 4, is factorial (Dolgachev, 1980, 1981 [4,5]) (for any field k). We prove a sufficient condition for the ring associated to a very general complex Laurent polynomial in d= 3 variables to be Q-factorial. © 2011 Elsevier B.V

Bruzzo, Ugo

Grassi, Antonella

Archivio istituzionale della ricerca - Alma Mater Studiorum Università di Bologna

This item was downloaded from IRIS Università di Bologna (https://cris.unibo.it/) When citing, please refer to the published version.       This is the final peer-reviewed accepted manuscript of:  Bruzzo, U., and A. Grassi. "Q-Factorial Laurent Rings." Journal of Pure and Applied Algebra, vol. 216, no. 4, 2012, pp. 894-896. The final published version is available online at: https://dx.doi.org/10.1016/j.jpaa.2011.10.016  Rights / License: The terms and conditions for the reuse of this version of the manuscript are specified in the publishing policy. For all terms of use and more information see the publisher's website.    Q-FACTORIAL LAURENT RINGSUGO BRUZZO§†‡ AND ANTONELLA GRASSI¶§ Institut des Hautes Études Scientifiques, Le Bois-Marie,91440 Bures-sur-Yvette, France† Istituto Nazionale di Fisica Nucleare, Sezione di Trieste¶ Department of Mathematics, University of Pennsylvania,David Rittenhouse Laboratory, 209 S 33rd Street,Philadelphia, PA 19104, USAAbstract. Dolgachev proves that the ring naturally associated to a generic Laurentpolynomial in d variables, d ≥ 4, is factorial [4, 5] (for any field k). We prove a sufficientcondition for the ring associated to a very general complex Laurent polynomial in d = 3variables to be Q-factorial.1. IntroductionIn [4] and Dolgachev [5] proves that the ring AF naturally associated to generic Laurentpolynomial F in d variables, d ≥ 4, with coefficients in any field k, is factorial. The basicingredient in Dolgachev’s proof is Grothendieck’s Lefschetz-type theorem ([6], Prop. 3.12)which, among other things, shows that under suitable conditions, the natural restrictionmap Pic(X) → Pic(Y ), where X is a scheme and Y is subvariety corresponding to an idealsheaf in OX , is an isomorphism. This result can be applied only when d ≥ 4.Date: November 7, 2018.1991 Mathematics Subject Classification. 16S34, 14J70, 14M25.E-mail: bruzzo@sissa.it, grassi@sas.upenn.edu.‡ On leave of absence from Scuola Internazionale Superiore di Studi Avanzati, Via Bonomea 265, 34136Trieste, ItalySupport for this work was provided by the NSF Research Training Group Grant DMS-0636606, by prin“Geometria delle varietà algebriche e dei loro spazi dei moduli ” and the infn project pi14 “Nonpertur-bative dynamics of gauge theories”. U.B. is a member of the vbac group.12 Q-FACTORIAL LAURENT RINGSIn this paper we consider the case d = 3, assuming that k = C, and prove a sufficientcondition for the ring AF to be Q-factorial (Theorem 3.1). The proof of this fact followsthe lines of Dolgachev’s proof, with Grothendieck’s result replaced by a Noether-Lefschetztheorem for hypersurfaces in toric 3-folds (Theorem 2.5) that we proved in [2].Acknowledgement. We thank Igor Dolgachev for interesting correspondence leading tothis result and the referee for useful comments. The authors are grateful for the hospitalityand support offered by the University of Pennsylvania, SISSA and IHES.2. PreliminariesWe follow the notation in [1] and [2]. Let M be a d-dimensional lattice, N = Hom(M,Z)and TN = N ⊗ C∗ the associated algebraic torus. Let Σ ⊂ NR be a complete simplicialfan, and denote by XΣ the corresponding complete toric variety. The torus TN naturallyacts on XΣ; Tτ ⊂ XΣ denotes the orbit of a subset of XΣ corresponding to a face τ of Σunder this action; the open dense orbit is denoted by T0.Definition 2.1. [1, Def. 4.13] A hypersurface X in XΣ is nondegenerate if X ∩ Tτ is asmooth 1-codimensional subvariety of Tτ for all faces τ in Σ.XΣ has only abelian quotient singularities, and is therefore an orbifold.Proposition 2.2. [1, Prop. 3.5, 4.15] Let L be a ample line bundle on XΣ. The hypersurfaceX ⊂ XΣ given by the zero locus of a generic section of L is nondegenerate. Moreover, Xis an orbifold.Since X is an orbifold, its complex cohomology has a pure Hodge structure [9]. This isan essential point in the proof of our Theorem 2.5.Definition 2.3 (The Cox Ring [3]). Consider a variable zi for each 1-dimensional coneςi, i = 1, . . . , n in Σ, and let S(Σ) be the polynomial ring C[z1, . . . , zn].The Cox ring has a natural gradation given by its class group Cl(Σ) of XΣ.Let L be an ample line bundle on XΣ, and let f ∈ H0(XΣ, L) ≃ S(Σ)β, where β =deg(L).Definition 2.4. The Jacobian ring of f is the quotient R(f) = S(Σ)/J(f), where J(f) isthe ideal in S(Σ) generated by the derivatives of f .Q-FACTORIAL LAURENT RINGS 3The Jacobian ring R(f) inherits a natural gradation from S(Σ).The next theorem was proved in [2], and will be key to proving our result about Laurentrings. We assume d = 3. We recall that the Picard number is defined as the rank of theclass group.Theorem 2.5. [2] Let XΣ a complete simplicial toric variety, and X ⊂ XΣ a very generalhypersurface cut by a section f of an ample line bundle L such that the multiplicationmorphismR(f)β ⊗ R(f)β−β0 → R(f)2β−β0 (1)is surjective (here β = deg(L) and β0 = − deg(KXΣ), where KXΣ is the canonical sheaf ofXΣ). Then X has the same Picard number as XΣ.Recall that a property is very general if it holds in the complement of countably manyproper subvarieties.If X is a quartic surface in P3, or more generally a K3 surface defined by a section ofthe anticanonical divisor in a simplicial toric variety, then the above map is surjective [2].It is a classical result that the map is not surjective if X is a cubic in P3.3. Q-factorial Laurent ringsThe ring C[M ] may be identified with the ring of regular functions on the torus TN ≃T0 ⊂ XΣ. An element F ∈ C[M ] is called a Laurent polynomial; F may be regarded as asection of the ample line bundle L, and it defines a hypersurface XF in XΣ.Let ∆ ⊂ M ⊗Z R be the polytope uniquely determined by the fan Σ and L (see [8],Lemma 2.14). To each Laurent polynomial F on can associate a polytope ∆F , called theNewton polytope of F . This is most easily described by choosing an isomorphism M ≃ Zd,writingF =∑i1,...,id∈Zdai1,...,id ti11· · · tiddand definingsupp(F ) = {i1, . . . , id ∈ Zd |ai1,...,id 6= 0}.∆F is then defined to be the convex hull of supp(F ) and Γ(∆) the set of all Laurentpolynomials such that ∆F ⊂ ∆. Γ(∆) is a finite dimensional vector space over C.By results given in [7] (see also [8], Chapter 2) a Laurent polynomial F extends to ameromorphic function on XΣ, which is a section of an ample line bundle LF . Thus, F4 Q-FACTORIAL LAURENT RINGSmay be regarded as an element in S(Σ)β, where β = deg(LF ). Denote by AF the ringC[M ]/(F ).Theorem 3.1. Let d = 3, and let F be a very general Laurent polynomial in Γ(∆); setβ = deg(LF ) and β0 = − deg(KXΣ). If the multiplication morphismR(F )β ⊗ R(F )β−β0 → R(F )2β−β0 (2)is surjective, the ring AF is Q-factorial.The proof that AF is Q-factorial follows closely the proof of Theorem 1.1 in [4]. Thebasic idea is to formulate the problem in a geometric way:Proof. Let XF ⊂ XΣ be the hypersurface cut by F (as a section of LF ). By Proposition2.2 the hypersurface XF is nondegenerate, and is an orbifold.Note that the ring AF may be identified with the ring of regular functions on the affinepart UF = XF ∩T0 of XF . Since the Picard group of T0 is trivial, every Cartier divisor inXΣ is linearly equivalent to a divisor supported in XΣ −T0. By Theorem 2.5, XF has thesame Picard number asXΣ, i.e., ρ(XF ) = ρ(XΣ). Then any Cartier divisor inXF is linearlyequivalent modulo torsion to a divisor supported in XF − UF , so that Pic(UF ) ⊗ Q = 0.Since UF is normal (actually smooth), then Cl(UF )⊗Q = 0. As UF ≃ Spec(AF ), we haveCl(AF )⊗Q = 0. References[1] V. V. Batyrev and D. A. Cox, On the Hodge structure of projective hypersurfaces in toric varieties,Duke Math. J., 75 (1994), pp. 293–338.[2] U. Bruzzo and A. Grassi, Picard group of hypersurfaces in toric varieties. arXiv:1011.1003[math.AG].[3] D. A. Cox, The homogeneous coordinate ring of a toric variety, J. Algebraic Geom., 4 (1995), pp. 17–50.[4] I. Dolgachev, Newton polyhedra and factorial rings, J. Pure Appl. Algebra, 18 (1980), pp. 253–258.Erratum, 21 (1981), pp. 9–10.[5] , Erratum: Newton polyhedra and factorial rings, J. Pure Appl. Algebra, 21 (1981), pp. 9–10.[6] A. Grothendieck, Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux etglobaux (SGA 2), Exposé XI, Documents Mathématiques (Paris), 4, Société Mathématique de France,Paris, 2005, pp. x+208. Séminaire de Géométrie Algébrique du Bois-Marie, 1962.[7] A. G. Khovanskĭı, Newton polyhedra and toroidal varieties, Funct. Anal. Appl., 11 (1977), pp. 289–296.Q-FACTORIAL LAURENT RINGS 5[8] T. Oda, Convex bodies and algebraic geometry. An introduction to the theory of toric varieties, vol. 15of Ergebnisse der Mathematik und ihrer Grenzgebiete (3), Springer-Verlag, Berlin, 1988.[9] M. Saito, Mixed Hodge modules, Publ. Res. Inst. Math. Sci., 26 (1990), pp. 221–333.

Dolgachev proves that the ring naturally associated to a generic   Laurent polynomial in d variables, d<4, is  factorial (for any  field k). We prove a sufficient condition for the ring associated to a very general complex Laurent polynomial in d=3 variables to be Q-factorial

BRUZZO U

GRASSI A

SISSA Digital Library

Q-factorial Laurent rings

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