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Regularity and Segre-Veronese Embeddings PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 137, Number 6, June 2009, Pages 1883–1890 S 0002-9939(09)09783-4 Article electronically published on January 15, 2009 REGULARITY AND SEGRE-VERONESE EMBEDDINGS DAVID A. COX AND EVGENY MATEROV (Communicated by Bernd Ulrich) Abstract. This paper studies the regularity of certain coherent sheaves that arise naturally from Segre-Veronese embeddings of a product of projective spaces. We give an explicit formula for the regularity of these sheaves and show that their regularity is subadditive. We then apply our results to study the Tate resolutions of these sheaves. 1. Regularity We will use the following generalized concept of regularity. Definition 1.1. Fix a projective variety X and line bundles L, B on X such that B is generated by global sections. Then a coherent sheaf F on X is L-regular with respect to B provided that Hi(X, F⊗L ⊗ B⊗(−i)) = 0 for all i>0. Here B⊗(−i) denotes the dual of B⊗i. This notion of regularity is a special case the multigraded regularity for sheaves introduced in [6], which in turn is a modification of the regularity defined in [8] for multigraded modules over a polynomial ring. We can relate Definition 1.1 to the Castelnuovo-Mumford regularity of a sheaf on projective space as follows. Suppose that B is very ample and L = i∗F,where i : X → PN is the projective embedding given by B. Then, given an integer p,the isomorphism Hi(PN , L(p − i)) Hi(X, F⊗B⊗p ⊗ B⊗(−i)) shows that F is B⊗p-regular with respect to B if and only if L is p-regular as a coherent sheaf on PN . Now fix r-tuples l := (l1,...,lr)andd := (d1,...,dr) of positive integers. These give the product variety X := Pl1 ×···×Plr r of dimension n := li and the d-uple Segre-Veronese embedding i=1 N r lk+dk νd : X −→ P ,N= − 1. k=1 dk The r-tuples l and d will be fixed for the remainder of the paper. Received by the editors May 29, 2008. 2000 Mathematics Subject Classification. Primary 13D02; Secondary 14F05, 16E05. Key words and phrases. Regularity, Segre-Veronese embedding, Tate resolution. c 2009 American Mathematical Society Reverts to public domain 28 years from publication 1883 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1884 DAVID A. COX AND EVGENY MATEROV Given m := (m1,...,mr), consider the line bundle OX (m)=OX (m1,...,mr) defined by O ∗O ⊗···⊗ ∗O X (m):=p1 Pl1 (m1) pr Plr (mr), l l l where pi : P 1 ×···×P r → P i is the projection. In this paper, we will study the regularity (in the sense of Definition 1.1) of OX (m) with respect to the line bundles L := OX (p)=OX (p1,...,pr),pi ∈ Z, B := OX (d)=OX (d1,...,dr). ⊆{ } For any nonempty subset J 1,...,r let lJ denote the sum j∈J lj.Hereisour first main result. Theorem 1.2. Let L = OX (p) and B = OX (d) be as above. Then OX (m) is L-regular with respect to B if and only if (1.1) max pk + mk + lk − lJ dk ≥ 0 k∈J for all nonempty subsets J ⊆{1,...,r}. Proof. Observe that i ⊗(−i) i (1.2) H (X, OX (m) ⊗ L ⊗ B )=H (X, OX (m + p − id)) and also that r i O − ik Plk O − H (X, X (m + p id)) = H ( , Plk (mk + pk idk)) i1+···+ir =i k=1 ik Plk O by the K¨unneth formula. Since H ( , Plk (j)) = 0 when ik =0,lk,wemay assume that i = l for some ∅ = J ⊆{1,...,r} and that J lk,k∈ J, ik = 0,k/∈ J. First suppose that (1.1) is satisfied for all J = ∅.GivensuchaJ,pickk ∈ J such that pk + mk + lk − lJ dk ≥ 0. − ≥− lk Plk O − Then mk +pk lJ dk lk,sothatH ( , Plk (mk +pk lJ dk)) = 0 by standard vanishing theorems for line bundles on projective space. By the above analysis, it follows easily that (1.2) vanishes for i>0. Next suppose that (1.3) max pk + mk + lk − lJ dk < 0 k∈J for some J = ∅. Among all such subsets J, pick one of maximum cardinality. For this J, we will prove that (1.2) is nonzero when i = lJ .BytheK¨unneth formula, it suffices to show that lk Plk O − ∈ H ( , Plk (mk + pk lJ dk)) =0,kJ, (1.4) 0 Plk O − ∈ H ( , Plk (mk + pk lJ dk)) =0,k/J. The first line of (1.4) is easy, since by Serre duality, ∗ lk Plk O − 0 Plk O − − − − H ( , Plk (mk + pk lJ dk)) H ( , Plk ( mk pk + lJ dk lk 1)) . License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use REGULARITY AND SEGRE-VERONESE EMBEDDINGS 1885 This is nonzero provided −mk − pk + lJ dk − lk − 1 ≥ 0, which for k ∈ J follows immediately from (1.3). For the second line of (1.4), suppose that k/∈ J.Bythe maximality of J,wemusthave (1.5) max ps + ms + ls − lJ∪{k}ds ≥ 0. s∈J∪{k} Note that lJ∪{k} = lJ + lk. If the maximum in (1.5) occurs at an element of J,say s ∈ J,then ps + ms + ls − (lJ + lk)ds ≥ 0, which is impossible since s ∈ J and (1.3) imply that ps + ms + ls − IJ ds < 0. Hence the maximum occurs at k,sothat pk + mk + lk − (lJ + lk)dk ≥ 0. This implies pk + mk − lJ dk ≥ lkdk − lk ≥ 0. It follows that the second line of (1.4) is nonzero, as desired. r If we fix the sheaf OX (m)andletL = OX (p)varyoverallp ∈ Z ,wegetthe regularity set r Reg(OX (m)) = {p ∈ Z |OX (m)isOX (p)-regular for B = OX (d)}. This set is easy to describe. Given a permutation σ in the symmetric group Sr,let J(σ, k) ⊆{1,...,r} denote the subset J(σ, k)={σ(i) | i ≥ σ−1(k)}, 1 ≤ k ≤ r. Thus J(σ, σ(1)) = {σ(1),σ(2),...,σ(r)} J(σ, σ(2)) = {σ(2),...,σ(r)} . J(σ, σ(r)) = {σ(r)}. Then define pσ =(−m1 − l1 + lJ(σ,1)d1,...,−mr − lr + lJ(σ,r)dr). r Proposition 1.3. The regularity set of OX (m) is the union of r! translates of N given by r Reg(OX (m)) = pσ + N . σ∈Sr Proof. First suppose that OX (m)isOX (p)-regular for B. We build a permutation σ ∈ Sr as follows. Theorem 1.2 tells us that (1.1) holds for all J = ∅. Hence, for J = {1,...,r},thereisk1 such that − ≥ pk1 + mk1 + lk1 lJ dk1 0. Setting σ(1) = k1, this becomes ≥− − pk1 mk1 lk1 + lJ(σ,k1)dk1 . Applying (1.1) to J = {1,...,r}\{σ(1)} gives k2 = k1 with − ≥ pk2 + mk2 + lk2 lJ dk2 0. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1886 DAVID A. COX AND EVGENY MATEROV Setting σ(1) = k2, this becomes ≥− − pk2 mk2 lk2 + lJ(σ,k2)dk2 . r Continuing in this way gives σ ∈ Sr with p ∈ pσ + N . r Conversely, let p ∈ pσ + N and take any nonempty J ⊆{1,...,r}.Write J = {σ(i1),...,σ(it)} for i1 < ···<it.Thensetk = σ(i1) ∈ J and observe that J ⊆{σ(i1),σ(i1 +1),...,σ(r)} = J(σ, σ(i1)) = J(σ, k). r This implies that lJ ≤ lJ(σ,k).Sincep ∈ pσ + N ,weobtain pk ≥−mk − lk + lJ(σ,k)dk ≥−mk − lk + lJ dk, so that pk + mk + lk − lJ dk ≥ 0. This proves that (1.1) holds for J, and since J = ∅ was arbitrary, we have p ∈ Reg(OX (m)) by Theorem 1.2. We next apply our results to study the Castelnuovo-Mumford regularity of OX (m) under the projective embedding given by the Segre-Veronese map l1 lr N νd : P ×···×P −→ P N coming from OX (d). This gives the sheaf F(m):=νd∗OX (m)onP . N Theorem 1.4. The sheaf F(m)=νd∗OX (m) is p-regular on P if and only if mk + lk p ≥ max min lJ − . J= ∅ k∈J dk Hence the regularity of F(m) is given by mk + lk reg(F(m)) = max min lJ − . J= ∅ k∈J dk Proof. Let B = OX (d). As noted earlier, F(m)isp-regular if and only if OX (m) ⊗p ⊗p is B -regular with respect to B.SinceB = OX (pd), Theorem 1.2 implies that F(m)isp-regular if and only if max pdk + mk + lk − lJ dk ≥ 0 k∈J for all subsets J = ∅ of {1,...,r}.Thisisequivalentto mk + lk p ≥ min lJ − k∈J dk for all J = ∅. From here, the first assertion of the theorem follows easily, and the second assertion is immediate. Example 1.5. When r = 2, the usual Segre embedding ν : Pa × Pb → Pab+a+b gives F(k, l)=ν∗OPa×Pb (k, l). Since d =(1, 1), Theorem 1.4 implies that reg(F(k, l)) = max{a − k + a ,b− l + b , min{a + b − k + a ,a+ b − l + b }} =max{−k, −l, min{b − k, a − l}} =max{− min{k, l}, min{b − k, a − l}}. This is the regularity formula from [4, Lem. 3.1]. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use REGULARITY AND SEGRE-VERONESE EMBEDDINGS 1887 N Example 1.6.
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