Syzygy Bundles of Non-complete Linear Systems: Stability and Rigidness

Let (X, L) be a polarized smooth projective variety. For any basepoint-free linear system LV\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {L}_{V}$$\end{document} with V⊂H0(X,OX(L))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V\subset {{\,\textrm{H}\,}}^{0}(X,\mathcal {O}_{X}(L))$$\end{document}, we consider the syzygy bundle MV\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_{V}$$\end{document} as the kernel of the evaluation map V⊗OX→OX(L)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V\otimes \mathcal {O}_{X}\rightarrow \mathcal {O}_{X}(L)$$\end{document}. The purpose of this article is twofold. First, we assume that MV\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_{V}$$\end{document} is L-stable and prove that, in a wide family of projective varieties, it represents a smooth point [MV]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[M_{V}]$$\end{document} in the corresponding moduli space M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {M}$$\end{document}. We compute the dimension of the irreducible component of M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {M}$$\end{document} passing through [MV]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[M_{V}]$$\end{document} and whether it is an isolated point. It turns out that the rigidness of [MV]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[M_{V}]$$\end{document} is closely related to the completeness of the linear system LV\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {L}_{V}$$\end{document}. In the second part of the paper, we address a question posed by Brenner regarding the stability of MV\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_{V}$$\end{document} when V is general enough. We answer this question for a large family of polarizations of X=Pm×Pn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X=\mathbb {P}^{m}\times \mathbb {P}^{n}$$\end{document}.


Introduction
Given a smooth projective variety X and a very ample line bundle L on X, let V ⊂ H 0 (X, O X (L)) be a subspace such that the corresponding linear system L V is basepoint-free.The projective morphism φ V : X −→ P(V * ) is a central object of study in algebraic geometry.In particular, the syzygies of φ V are encoded in the so-called syzygy bundle M V , whose study has been of increasing interest in the last decades.Namely, we define the syzygy bundle M V as the kernel of the evaluation map ev : V ⊗O X → O X (L), which is surjective.In particular we have the following short exact sequence which describes M V as a vector bundle of rank dim V − 1.The syzygy bundles M V have been studied from many perspectives in the last decades.In particular when V = H 0 (X, O X (L)), the linear system L V is complete and the morphism φ V coincides with the embedding φ L : X ֒→ P(H 0 (X, O X (L)) * ), given by the very ample line bundle L. In this case, the syzygy bundle is denoted by M L := M V , and it is behind many geometric properties of the embedding φ L .For instance, the properties (N p ) in the sense of Green [14], or the stability of the pullback φ * L T P N L of the tangent bundle of P N L , which is related to the stability of M L and has been studied thoroughly in the recent years [2,7,8,15,17,18,19] and has led to the so-called Ein-Lazarsfeld-Mustopa Conjecture (see Conjecture 4.14).In this work, we focus our attention on the stability of the syzygy bundles M V arising from linear systems L V which are non-necessarily complete.
In [1], motivated by the theory of tight closure, the systematic study of L-stable syzygy bundles on P n was considered.In particular, in [1,Question 7.8] Brenner asked the following question: Question 1.1.Let us consider integers n, d ≥ 1.For which integers r such that n + 1 ≤ r ≤ n+d d there exist r monomials m 1 , . . ., m r with no common factors such that the syzygy bundle M V corresponding to the subspace V = m 1 , . . ., m r ⊂ H 0 (P n , O P n (d)) is semistable?
The case r = n+d n had been previously proved in [12], and a complete answer was given in [6] and [3].Moreover, in [6] the authors studied the local geometry of the moduli space M in which an L-stable syzygy bundle may be represented.In [6,Theorem 4.4] they proved that apart from few exceptions, an L−stable syzygy bundle M V correspond to a smooth point in M and they computed the dimension of the irreducible component containing it.From their result one can see that if n ≥ 4, then M V is infinitesimally rigid if and only if r = n+d n and the linear system is complete.As we show in Theorem 3.1, this is not a particular feature of syzygy bundles on projective spaces, and we can generalize this fact to a large family of smooth projective varieties.
In this paper, we consider the analogous of Brenner's Question 1.1 for any smooth projective variety: Question 1.2.Let us fix an ample line bundle L on a smooth projective variety X of dimension d.For which integers dim(X) + 1 ≤ r ≤ dim H 0 (X, L), is there a basepoint-free linear system L V associated to a subspace V ⊂ H 0 (X, L) of dimension r, such that the syzygy bundle M V is L−stable?
Since stability is an open property, we notice that positively answering Question 1.2 for a certain integer r we obtain that the syzygy bundle M V corresponding to a general subspace V ⊂ H 0 (X, O X (L)) of dimension r is L-stable.Notwithstanding, the implications of Question 1.2 go beyond this fact: it sheds new light on the geometry of certain moduli spaces of L-stable vector bundles on a projective variety.To be more precise, in Section 3 we consider a large family of polarized smooth projective varieties (X, L) of any dimension.In this setting, we show (see Theorem 3.1) that a positive answer to Question 1.2 automatically yields smooth points on a suitable moduli space.Even more, we show that in this case the dimension of the irreducible component containing these points is fully described by only using the very ample line bundle L. It is worthwhile to mention that this family of projective varieties include smooth complete projective toric varieties, Grassmannians, flag varieties among others.On the other hand, Theorem 3.1 works under well understood assumptions on the very ample line bundle L, which are actually very mild hypothesis when dim(X) = 3.
Motivated by these facts, we devote the second half of this paper to answer Question 1.2 for the product of two projective spaces X = P m × P n , which is an example of a smooth complete projective toric variety.In this case, we can use the Cox ring of X which is the standard-bigraded polynomial ring K[x 0 , . . ., x m , y 0 , . . ., y n ] and we examine the possible degrees of syzygies among r forms {f 1 , . . ., f r } of degree (a, b) with a, b > 0. This allows us to give a positive answer to Question 1.2 in a large amount of cases (see Theorem 4.7).
This work is organized as follows.In Section 2 we gather the basic results regarding stability of vector bundles on polarized projective varieties (X, L) and the theory of syzygy bundles of linear systems needed in the sequel.Afterwards, the paper is divided in two main sections.In Section 3 we consider a large family of polarized projective varieties (X, L), which include, but is not limited to, smooth complete projective toric varieties, Grassmannians or flag varieties.We prove (Theorem 3.1) that in this setting an L-stable syzygy bundle of a non-necessarily complete linear system corresponds to a smooth point in its moduli space, and we give explicitly the dimension of the irreducible component containing that point.The second part of this work is found in Section 4, where we aim to answer Question 1.2 for products of projective spaces.Our main results in this regard are Theorem 4.7 and Corollary 4.10 which answers Question 1.2 in a large number of cases.In Theorem 4.12 we apply this results to give insight on the moduli spaces of syzygy bundles on P m × P n .Finally, we end this section posing some open questions regarding the stability of syzygy bundles of non-complete linear systems.

Basic results
Let (X, L) be a polarized smooth projective variety of dimension d, defined over an algebraically closed field K of characteristic zero and let L be a globally generated line bundle.For any vector subspace V ⊂ H 0 (X, L) we denote by L V the corresponding linear system.If L V is base point-free, we define the syzygy bundle M V as the kernel of the evaluation map Notice that M V is a vector bundle fitting in the following short exact sequence (1) 0 In particular we have: The goal of this paper is to study the stability of the vector bundle M V for appropriate subspaces V ⊂ H 0 (X, O X (L)) and to obtain local information on the geometry of their corresponding moduli spaces.Let us first recall the definition and some key result about the stability of vector bundles.Definition 2.1.Let (X, L) be a polarized smooth variety of dimension d.A vector bundle E on X is L−stable (resp.L−semistable) if for any subsheaf F ⊂ E with 0 < rk(F ) < rk(E), we have The following result is a cohomological characterization of the stability, and it will play a central role in the proof of our main result.Lemma 2.2.[3, Lemma 2.1] Let (X, L) be a polarized smooth variety of dimension d.Let E be a vector bundle on X. Suppose that for any integer q and any line bundle G on X such that Remark 2.3.A vector bundle E satisfying the hypothesis of Lemma 2.2 is said to be cohomologically stable.It is worthwhile to point out that any cohomological stable vector bundle on a polarized variety (X, L) is L-stable but not vice versa.
The stability of syzygy bundles associated to complete linear systems (i.e. when V = H 0 (X, O X (L))) on polarized varieties (X, L) has received a lot of attention on the last decades (see, for instance, [2,7,8,12,18,19]).Our goal is to answer the following much more general question: Question 2.4.Let us fix an ample line bundle L on a smooth projective variety X of dimension d.For which integers r ≤ dim H 0 (X, L), is there a base point-free linear system L V associated to a subspace V ⊂ H 0 (X, L) of dimension r, such that the syzygy bundle M V is L−stable?Question 2.4 is a generalization of a question raised by Brenner in [1, Question 7.8], regarding the stability of syzygy bundles of non-complete linear systems in P N .This problem has been further studied in [6,16,3], where a complete answer for the case (X, L) = (P N , O P N (d)) is given.Remark 2.5.i) Since the L−stability is an open property, Question 2.4 is equivalent to ask for which integers r ≤ H 0 (X, L), the syzygy bundle M V corresponding to a general base point-free linear system L V given by a subspace V ⊂ H 0 (X, L) of dimension r, is L−stable.
ii) In the case V = H 0 (X, L) there is a conjecture by Ein, Lazasferld and Mustopa (see [8,Conjecture 2.6] or Conjecture 4.14) which addresses the stability of the syzygy bundle M V .
Remark 2.6.As shown in Section 3, answering Question 2.4, and thus providing general syzygy bundles which are L−stable, shed new light on the geometry of the moduli spaces where the syzygy bundles are represented as a point.

Rigidness of the syzygy bundles
In this section we focus on a polarized projective variety (X, L) of dimension d and we consider a syzygy bundle M V associated to a base point-free linear system In this section we assume that M V is L-stable, and we study the geometry of this moduli space M around [M V ].
Recall that the Zariski tangent space of M at a point [E] is canonically given by and we say that We have the following result: In particular, when dim(X) ≥ 3, M V is infinitesimally rigid if and only if V = H 0 (X, O X (L)).
Proof.Let us start studying H 2 (X, Dualizing the exact sequence (2) and tensoring it by M V , we obtain: From the exact sequence of cohomology of (2) and the hypothesis H We leave Case b) to the end of the proof.In any other case we have H 3 (X, O X (−L)) = 0 and it follows that From (3) and using (5) we have that Finally, from (2) we have that Finally, from (9) and using (10), (11) and (13) we get: In particular, for Case d) (dim(X) = 2), the proof is finished.
On the other hand, notice that if dim(X) ≥ 3, by Kodaira's vanishing theorem we have H 2 (X, O X (−L)) = 0. Thus, we have To finish the proof we need to tackle Case b), that is when dim(X) = 3 and V = H 0 (X, O X ).In this case we have not seen that [M V ] is a smooth point in M so we cannot deduce directly that dim K T [M V ] gives the dimension of the irreducible component of M containing [M V ].However, from (14) we obtain that ii) On the other hand, when dim(X) = 3 the condition H 3 (X, O X (−L)) = 0 is not always satisfied even in the case of complete smooth toric varieties, as the following example shows: take X = P 3 and L = O P 3 (i) with i ≥ 4. When X is a complete toric variety of dimension n, this technical condition can be tackled using the Batirev-Borisov vanishing theorem.Indeed, for any Cartier nef divisor D on X there is a lattice polytope P D such that where |P D | is the number of lattice points of P D and | Relint(P D )| is the number of lattice points in the relative interior of the polytope P D (see [5, Section 5 and Theorem 9.2.7]).Theorem 3.1 has been recently generalized in [10] and [11] where using generalized syzygy bundles we construct recursively open subspaces of moduli spaces of simple sheaves on X that are smooth, rational, quasiprojective varieties.
At the end of Section 4 we will apply the results of this section (see Theorem 4.12) to general syzygy bundles on P m × P n associated to linear systems.

Stability of syzygy bundles of non-complete linear systems
The aim of this Section is to answer Question 2.4 for X = P m × P n .Notice that X may be viewed as the smooth complete toric variety toric variety with Cox ring Any line bundle on X is of the form O X (a, b) for some integers a, b.A line bundle O X (a, b) is ample if and only if it is very ample, if and only if a, b > 0; and it is effective if and only if a, b ≥ 0.Moreover, we have the following identification of vector spaces: In this setting, Lemma 2.2 can be rephrased as follows: Lemma 4.1.Take X = P m × P n and L = O X (a, b) a very ample line bundle.Let V ⊂ H 0 (X, L) be a vector space such that L V is a base point-free linear system.The syzygy bundle M V is L−stable if for any 0 < q < r − 1 and any line bundle G = O X (x, y), then Proof.It follows from Lemma 2.2 using that and Notice that if G = O(x, y) is a line bundle we have, from (1), the following inclusion of vector spaces Hence, if H 0 (X, q M V (x, y)) = 0, then G = O X (x, y) is effective.Therefore, we have x, y ≥ 0.
Our first goal is to prove that the syzygy bundle M V associated to any sufficiently large vector space V ⊂ H 0 (X, L) is always L−stable.More precisely, we show that for any base point-free linear system L V associated to an r−dimensional vector space V ⊂ H 0 (X, L) such that (15) a(m the syzygy bundle M V is L−stable.To this end, the following Lemma is needed. Lemma 4.2.Take X = P m × P n and L = O X (a, b) a very ample line bundle.Let V ⊂ H 0 (X, L) be a vector space such that L V is a base point-free linear system.For any 0 < q < r − 1 and any line bundle G = O X (x, y), if H 0 ( q M V (x, y)) = 0, then x + y ≥ q.
Proof.We have that M V (−L) = K V , where K V = syz(f 1 , . . ., f r ) is the syzygy module of the forms {f 1 , . . ., f r } ⊂ R (a,b) corresponding to a basis of V .Then, K V lies in the following exact sequence: In particular, if is a minimal presentation of K V , then we have for any 1 ≤ i ≤ µ that As a consequence, we have a minimal presentation ( 16) Taking exterior powers in (16) we have ( 17) Notice that for any q−uple 1 ≤ i 1 < . . .< i q ≤ µ it holds that Thus, if H 0 (X, q M V (x, y)) = 0, then we have that x + y ≥ q.
Proposition 4.3.Take X = P m × P n and L = O X (a, b) a very ample line bundle such that a ≥ b.Let L V be any basepoint-free linear system associated to a vector subspace V ⊂ H 0 (X, L) with then the syzygy bundle M V associated to L V is L−stable.
Proof.We apply Lemma 4.1.Let us consider an integer 0 < q < r − 1 and G = O X (x, y) a line bundle such that H 0 (X, q M V (x, y)) = 0. Since x, y ≥ 0 and a ≥ b, we have that bmx + any ≥ b min(m, n)(x + y).Therefore, it is enough to see that it holds bnx + amy > qab(m + n) r − 1 .
Since H 0 (X, q M V (x, y)) = 0 then we have that there is some q−uple (i 1 , . . ., i q ) with 1 ≥0 .Thus, we have that x + y ≥ q.Consequently, The remaining of this section is devoted to show that for any integer r such that ( 18) there is a vector subspace V ⊂ H 0 (X, L) with dim(V ) = r such that the associated linear system L V is base point-free and the corresponding syzygy bundle M V is L−stable.
Notation 4.4.For any integer r satisfying ( 18), we denote by t r the only integer such that Notice that it holds 2 ≤ t r ≤ b.
The following lemma is in the core of the proof.
In particular, condition ii) implies that the linear system L W associated to W is base point-free.Case A).We consider the following two sets of monomials in R (a,b) : We construct the vector space W from the sets of monomials A and B, fulfilling conditions i), ii) and iii).We consider two subcases: We have dim W = 2⌊ a tr −1 ⌋ + 3. Otherwise, a tr−1 ∈ Z is even, and we define We have dim W = 2⌊ a tr −1 ⌋ + 2. Otherwise, a tr−1 ∈ Z is even, and we define We have dim W = 2⌊ a tr −1 ⌋ + 1.In any case, we have that dim W ≥ 2a tr−1 + 1 and conditions i), ii) and iii) hold.Case B).We consider the following sets of monomials in R (a,b) for each 1 ≤ i ≤ m − 1: Otherwise, a tr−1 ∈ Z and it is even, and we define and we have dim In any case, we have dim W ≥ (m+1)a tr −1 + 1 and conditions i), ii) and iii) hold.Case C).We consider the following sets of monomials in R (a,b) for each 1 ≤ i ≤ n − 1: and we have dim Otherwise, a tr−1 ∈ Z and it is even, and we define and we have dim W = (n + 1)(⌊ a t r − 1 ⌋ − 2) + 2(n + 1).
In any case, we have dim W ≥ (n+1)a tr−1 + 1 and conditions i), ii) and iii) hold.Case D).Let us consider now the vector space W constructed in Case B) (respectively Case C)) taking the subring R ′ = K[x 0 , . . ., x m , y 0 , y 1 ] ⊂ R, if m = max(m, n) (respectively taking the subring R ′ = K[x 0 , x 1 , y 0 , . . ., y n ], if n = max(m, n)).Thus, W is also a vector subspace in R (a,b) and conditions i), ii) and iii) are automatically satisfied.On the other hand, we have that dim W ≥ a(max(m, n) + 1) Proposition 4.6.Take X = P m × P n and L = O X (a, b) a very ample line bundle, such that a ≥ b ≥ 2. For any integer let L V be general base point-free linear system associated to an r−dimensional vector space V ⊂ H 0 (X, L).Then, the syzygy bundle M V corresponding to L V is L−stable.
Proof.By Remark 2.5 it is enough to find an r−dimensional vector space V ⊂ H 0 (X, L) such that the linear system L V is base point-free and its corresponding syzygy bundle M V is L−stable.To this end we use Lemma 4.5.By Notation 4.4, let us consider the integer 2 ≤ t r ≤ b such that We consider the N −dimensional vector subspace W given by Lemma 4.5.We may write , where m i is a monomial for any 1 ≤ i ≤ N − (m + 1)(n + 1).Then we construct the following r−dimensional vector space . Notice that syz(f 1 , . . ., f r ) ⊂ syz(f 1 , . . ., f N ).Thus, applying condition ii) of Lemma 4.5 to any syzygy ξ = (g 1 , . . ., g r ) ∈ syz(V ) of degree (a + α, b + β), we have that In particular, if K V := syz(f 1 , . . ., f r ), we have the following minimal presentation On the other hand, since M V = K V , we have the following minimal presentation which yields, taking the q−th exterior power, the minimal presentation of q M V : Now, we apply Lemma 2.2.Let us consider an integer 0 < q < r −1 and a line bundle We want to see that it holds bmx + any > qab(m + n) r − 1 .
Since a ≥ b, we have that bmx + any ≥ b min(m, n)(x + y).On the other hand, by (19), assuming that H 0 (X, q M V (x, y)) = 0 we have that x + y ≥ qt.Consequently, we obtain that let L V be a general base point-free linear system associated to an r−dimensional vector space V ⊂ H 0 (X, L).Then, the syzygy bundle M V corresponding to L V is L−stable.
Proof.Exchanging the role of m and n in P m × P n if necessary, we may assume that a ≥ b.On the other hand, if we assume b = 1, then we have Therefore, the result follows directly from Proposition 4.3.
On the other hand, assume that a ≥ b ≥ 2.Then, the result follows from Proposition 4.3 if We have the following remark: Remark 4.8.(i) Let V ⊂ H 0 (X, O X (L)) be a vector space corresponding to a base point-free linear system L V , then we have that By Theorem 4.7, we have seen that when V is general and then, the corresponding syzygy bundle M V is L−stable.However, to solve completely Question 2.4, it remains open the case of a general vector space V such that (ii) On the other hand, if we assume in addition that V is generated by monomials, then being L V a base point-free linear system implies that However, we notice that the open range of cases expressed in (21) is smaller than that of (20).
The following corollaries shows that in some cases Theorem 4.7 solves already Question 2.4 for P m × P n .then the syzygy bundle M V corresponding to a general basepoint-free linear system L V , is L−stable.
In particular, if m, n ≥ 2, for any integer t ≥ 1 we set H t := O X (t, t).Then, the syzygy bundle M V associated to a general basepoint-free linear system L V with V ⊂ H 0 (X, H t ) is H t −stable.Notice that then, M V is also H 1 −stable.
Proof.In this case, we have Since there is no base point-free linear system associated to a vector space V satisfying (20), the result follows from Theorem 4.7.then the syzygy bundle M V corresponding to a general base point-free linear system L V given by a general vector space V generated by monomials, is L−stable.
Proof.In this case, we have Since there is no base point-free linear system associated to a vector space V generated by monomials satisfying (21), the result follows from Theorem 4.7.
the syzygy bundle M V is L−stable.
As an application of Theorem 3.1 to this setting, we address now the geometry of the moduli space M on which an L−stable syzygy bundle M V can be represented (see Section 3).

Lemma 4 . 5 .
Let R = K[x 0 , . . ., x m , y 0 , . . ., y n ] be the Cox ring of X = P m × P n , and let a ≥ b ≥ 2 be two integers.For any integer r such that
, and for i = n: If a tr−1 / ∈ Z or a tr −1 ∈ Z and it is odd, we define

Propositions 4 .
3 and 4.6 yield the main result of this note.Theorem 4.7.Let X = P m × P n and a, b ≥ 1 two integers and let L := O X (a, b) be a very ample line bundle on X.For any integer r such that max(a, b)

Corollary 4 . 9 .
Let X = P m × P n and a, b ≥ 1 two integers and let L := O X (a, b) be a very ample line bundle on X.If max(a, b) < min(m, n) min(a, b),

Corollary 4 . 10 .
Let X = P m × P n and a, b ≥ 1 two integers and let L := O X (a, b) be a very ample line bundle on X.If max(a, b) < (mn + m + n) min(m, n) m + n min(a, b),