A GEOMETRIC CRITERION FOR THE FINITE GENERATION OF THE COX RING OF PROJECTIVE SURFACES 2 1 0 BRENDADELAROSANAVARRO,MUSTAPHALAHYANE, 2 ISRAELMORENOMEJ´IA,ANDOSVALDOOSUNACASTRO n a J 8 1 Abstract. The aimis to giveageometric characterization of the finite generation of the Cox ringof anticanonical rational surfaces. Thischar- ] acterization is encoded in the finite generation of the effective monoid. G Furthermore, we prove that in the case of a smooth projective rational A surfacehaving anegative multipleofits canonical divisorwithonly two linearly independent global sections (e.g., an elliptic rational surface), . h the finite generation isequivalent to the factthat there areonlyafinite t numberofsmoothprojectiverationalcurvesofself-intersection−1. The a groundfieldisassumedtobealgebraicallyclosedofarbitrarycharacter- m istic. [ 1. Introduction 1 v In [10], Galindo and Monserrat characterize the smooth projective surfaces 4 Z defined over an algebraically closed field k with finitely generatedCox rings 9 (see the next paragraph for the definition) by means of the finiteness of the 6 set of integral curves on Z of negative self-intersection and the existence of a 3 . finitelygeneratedk−algebracontainingtwok−algebrasassociatednaturallyto 1 Z, see [10, Theorem 1, page 94]. The aim of this work is to give an equivalent 0 characterization of the finite generation of the Cox ring totally based on the 2 1 geometry of the surface and to apply the criterion to some classes of smooth : projectiverationalsurfaces,e.g. theanticanonicalones(i.e.,thoserationalsur- v i faces holding an effective anticanonical divisor) and the surfaces constructed X in [5], [6], [8] and [9]; establishing thus the geometric nature of our character- r ization. For some purely algebraic features of the Cox ring of a variety, see a [7]. Following Hu and Keel [15], the Cox (or the total coordinate) ring of a smooth projective variety V defined over an algebraically closed field k is the k−algebra defined as follows: Cox(V)= M H0(V,O(Ln11 ⊗...⊗Lnrr)). (n1,...,nr)∈Zr 2000 Mathematics Subject Classification. Primary14J26; Secondary14F17,14F05. 1 2 DELAROSANAVARRO, LAHYANE,MORENOMEJ´IA,ANDOSUNACASTRO Here (L ,...,L ) is a basis of the Z-module Pic(V) of classes of invertible 1 r sheaves on V modulo isomorphisms under the tensor product, and we have assumed that the linear and numerical equivalences on the group of Cartier divisors on V are the same, such assumption is satisfied for example for the smooth projective rational surfaces V. An interesting (but still) open problem is to classify theoretically and/or effectively and constructively all smooth projective rational surfaces S for whichthek−algebraCox(S)isfinitelygenerated. MasayoshiNagata(see[23]) showedthatthesurfaceZ obtainedbyblowingupoftheprojectiveplaneP2 at nine or more points in general position has an infinite number of (−1)−curves (see also [19], [16], [17], [20], [24], [22], [21] and [18] for cases when the points need not be in general position), consequently its Cox ring Cox(Z) is not finitely generated. Here a (−1)−curve on Z means a smooth projective curve on Z of self-intersection equal to −1. Note that in this example, the effective monoidM(Z)ofZ isalsonotfinitelygenerated,whereM(Z)standsfortheset of elements of the Picard group Pic(Z) of Z having at least a nonzero global section. InthispaperwemainlylookforthosesmoothprojectiverationalsurfacesS for which the finite generation of Cox(S) is equivalent to the finite generation of M(S). Our two main results, Theorems 1 and 6 below which are derived from Theorem 14, give a partial answer. By the way, we have been informed by a referee that in the characteristic zero case, Theorem 14 was obtained in [1] using a different approach. Theorem 1. LetS beasmoothprojectiverationalsurfacedefinedoveran algebraically closed field k of arbitrary characteristic such that the invertible sheaf associated to the divisor −K has a nonzero global section. S The following assertions are equivalent: (1) Cox(S) is finitely generated. (2) M(S) is finitely generated. (3) S has only a finite number of (−1)-curves and only a finite number of (−2)-curves. Here K denotes a canonical divisor on S. S Proof. It follows from Lemma 10 and Theorem 14 below. (cid:3) As consequences, the following two results hold: Corollary 2. The Cox ring ofa smoothprojective rationalsurface having a canonical divisor of self-intersection larger than or equal to zero is finitely generated if and only if the set of (−1)-curves is finite. Proof. Apply Theorem 1 and [20, Proposition 4.3 (a), page 9]. (cid:3) A GEOMETRIC CRITERION FOR THE FINITE GENERATION OF THE COX RING ... 3 Corollary 3. The Cox ring ofa smoothprojective rationalsurface having a canonical divisor of self-intersection larger than zero is finitely generated. Proof. Apply Theorem 1 and [20, Proposition 4.3 (a), page 9]. (cid:3) In particular, since a Del Pezzo surface is nothing but a blow up of the projective plane at r points with r ≤ 8, we recover the well known result, see [3]: Corollary 4. The Cox ring of a Del Pezzo surface is finitely generated. Corollary 5. The Cox ring ofa smoothprojective rationalsurface having anintegralcurvealgebraicallyequivalenttoananti-canonicaldivisorisfinitely generated if and only if the set of (−2)-curves is finite and spans a linear subspace in the Picard group of codimension one. Proof. The result follows from [11] and Theorem 1. (cid:3) Here is our second result: Theorem 6. LetZ beasmoothprojectiverationalsurfacedefinedoveran algebraically closed field k of arbitrary characteristic such that the invertible sheaf associatedto the divisor −rK has only two linearly independent global Z sections for some positive integer r. The following assertions are equivalent: (1) Cox(Z) is finitely generated. (2) The set of smooth projective rational curves of self-intersection −1 on Z is finite. Here K denotes a canonical divisor on Z. Z Proof. Apply Theorem14below andthe fact thatthe setof(−2)-curveson Z is finite. (cid:3) 2. Preliminaries 2.1. General Notions. Let S be a smooth projective surface defined over an algebraically closed field of arbitrary characteristic. A canonical divisor on S, respectively the Picard group of S will be denoted by K and Pic(S) respec- S tively. There is an intersection form on Pic(S) induced by the intersection of divisorson S, it will be denoted by a dot, that is, for x and y in Pic(S), x.y is the intersection number of x and y (see [14] and [2]). The following result known as the Riemann-Roch theorem for smooth pro- jective surfaces is stated using the Serre duality. Lemma 7. LetD be a divisorona smoothprojective surfaceS having an algebraicallyclosedfieldofarbitrarycharacteristicasa groundfield. Thenthe 4 DELAROSANAVARRO, LAHYANE,MORENOMEJ´IA,ANDOSUNACASTRO following equality holds: 1 h0(S,O (D))−h1(S,O (D))+h0(S,O (K −D))=1+p (S)+ (D2−D.K ). S S S S a S 2 O (D) (respectively, p (S)) being aninvertiblesheafassociatedcanonicallyto S a thedivisorD (respectively,thearithmeticgenusofS,thatisχ(O )−1,where S χ is the Euler characteristic function). Here we recall some standard results, see [12], [14] and [2]. A divisor class modulo linear equivalence x of a smooth projective surface S is effective, re- spectively numerically effective (nef in short) if an element of x is an effective, respectively numerically effective, divisor on S. Here a divisor D on S is nef if D.C ≥ 0 for every integral curve C on S. Now, we start with some proper- ties which follow from a successive iterations of blowing up closed points of a smooth projective rational surface. Lemma 8. Let π⋆ : NS(X) → NS(Y) be the natural group homo- morphism on N´eron-Severi groups induced by a given birational morphism π : Y → X of smooth projective rational surfaces. Then π⋆ is an injective intersection-formpreservingmapoffreeabeliangroupsoffiniterank. Further- more, it preserves the dimensions of cohomology groups, the effective divisor classes and the numerically effective divisor classes. Proof. See [13, Lemma II.1, page 1193]. (cid:3) Lemma 9. Let x be an element of the N´eron-Severi group NS(X) of a smoothprojectiverationalsurfaceX. The effectiveness or the the nefness of x impliesthenoneffectivenessofk −x,wherek denotestheelementofPic(X) X X which contains a canonical divisor on X. Moreover, the nefness of x implies also that the self-intersection of x is greater than or equal to zero. Proof. See [13, Lemma II.2, page 1193]. (cid:3) Thefollowingresultisalsoneeded. Werecallthata(−1)-curve,respectively a(−2)-curve,isasmoothrationalcurveofself-intersection−1,respectively−2. Lemma 10. The monoid of effective divisor classes modulo linear equiva- lence on a smooth projective rationalsurface X having an effective anticanon- ical divisor is finitely generated if and only if X has only a finite number of (−1)-curves and only a finite number of (−2)-curves. Proof. See [20, Corollary 4.2, page 109]. (cid:3) 2.2. Extremal Surfaces. Let Nef(S) denotes the set of nef elements in the Picard group Pic(S) of a smooth projective surface S, it has obviously an A GEOMETRIC CRITERION FOR THE FINITE GENERATION OF THE COX RING ... 5 algebraic structure as a monoid. We define two more submonoids Char(S) and [Char(S):Nef(S)] of Pic(S) (see [4] and [10]) as follows: Definition 11. With notation as above. (1) The characteristic monoid Char(S) of S is the set of elements x in Pic(S)suchthatthereexistsaneffectivedivisoronSwhoseassociated complete linear system is base point free and whose class in Pic(S) is equal to x. (2) The monoid of fractional base point free effective classes [Char(S) : Nef(S)] of S is the set of elements y in Pic(S) such that there exists a positive integer n with ny ∈Char(S). ThemainpropertiesthatweareinterestedinregardingChar(S)and[Char(S): Nef(S)] of a smmoth projective surface are the ones in the Lemma below. Their proofs are straightforward. Lemma 12. With notation as above, the followings hold: (1) Char(S) and [Char(S):Nef(S)] are submonoids of Nef(S). (2) Char(S)⊆[Char(S):Nef(S)]. Here we define the ingredient needed for our criterion: Definition 13. With notation as above, S is extremal if the monoid of fractionalbasepointfree effective classes[Char(S):Nef(S)]ismaximal,that is, if Nef(S)=[Char(S):Nef(S)]. 3. The Criterion Now we are able to state our geometric criterion: Theorem 14. Let S be a smooth projective surface defined over an alge- braically closed field of arbitrary characteristic. The following assertions are equivalents: (1) The Cox ring Cox(S) is finitely generated. (2) S satisfies the following two properties: i. S is extremal, and ii. the effective monoid M(S) of S is finitely generated. (3) S satisfies the following two properties: i. S is extremal, and ii. the nef monoid Nef(S) of S is finitely generated. Proof. By duality, it is obvious that (2) is equivalent to (3). Assume that the Cox ring Cox(S) of S is finitely generated, it follows that the effective monoid M(S) of S is finitely generated. On the other hand, if y is an effective element of Nef(S), then if y belongs to Char(S), we are done and if not we 6 DELAROSANAVARRO, LAHYANE,MORENOMEJ´IA,ANDOSUNACASTRO mayfindsomepositiveintegerssuchthatsy isanelementofChar(S). Hence [Char(S) : Nef(S)], i.e., S is extremal. Conversely, if S is extremal, and the nefmonoidNef(S)ofS isfinitely generated,thenitfollowsfrom[10]thatthe Cox ring of S is finitely generated. (cid:3) Acknowledgements The first three authors were supported by PAPIIT IN102008researchgrant and the projects CIC-UMSNH 2010 and 2011. References [1] M. Artebani, J. Hausen, A. Laface On Cox rings of K3 surfaces, Compos. Math. vol. 146(2010), no4.964-998. [2] W. Barth, C. Peters , A. Van de Ven. Compact Complex Surfaces. Berlin, Springer (1984). 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[23] M. Nagata, On rational surfaces, II, Memoirs of the College of Science, University of Kyoto,SeriesA33(1960), no.2,271–293. [24] J.Rosoff,Effectivedivisorclassesand blowings-upof P2,PacificJournalofMathemat- ics,Volume89,Number2(1980), pp.419–429. InstitutodeF´ısicayMatema´ticas(IFM),UniversidadMichoacanadeSanNicola´s de Hidalgo, Edificio C-3, Ciudad Universitaria. C. P. 58040 Morelia, Michoaca´n, M´exico E-mail address: [email protected] InstitutodeF´ısicayMatema´ticas(IFM),UniversidadMichoacanadeSanNicola´s de Hidalgo, Edificio C-3, Ciudad Universitaria. C. P. 58040 Morelia, Michoaca´n, M´exico E-mail address: [email protected] Institutode Matema´ticas,UniversidadNacionalAuto´nomade M´exico,A´reade la Investigacio´n Cient´ıfica, Circuito Exterior, Ciudad Universitaria, Coyoaca´n, C.P. 04510,M´exico D.F., M´exico E-mail address: [email protected] InstitutodeF´ısicayMatema´ticas(IFM),UniversidadMichoacanadeSanNicola´s de Hidalgo, Edificio C-3, Ciudad Universitaria. C. P. 58040 Morelia, Michoaca´n, M´exico E-mail address: [email protected]