Regularity and Segre-Veronese embeddings

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.


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 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 → P N is the projective embedding given by B. Then, given an integer p, the isomorphism H i (P N , L(p − i)) ≃ H i (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 P N . Now fix r-tuples l := (l 1 , . . . , l r ) and d := (d 1 , . . . , d r ) of positive integers. These give the product variety X := P l1 × · · · × P lr of dimension n := r i=1 l i and the d-uple Segre-Veronese embedding The r-tuples l and d will be fixed for the remainder of the paper. Given m := (m 1 , . . . , m r ), consider the line bundle where p i : P l1 × · · · × P lr → P li is the projection. In this paper, we will study the regularity (in the sense of Definition 1.1) of O X (m) with respect to the line bundles For any nonempty subset J ⊆ {1, . . . , r} let l J denote the sum j∈J l j . Here is our first main result.
Proof. Observe that and also that by the Künneth formula. Since H i k (P l k , O P l k (j)) = 0 when i k = 0, l k , we may assume that i = l J for some ∅ = J ⊆ {1, . . . , r} and that First suppose that (1.1) is satisfied for all J = ∅. Given such a J, pick k ∈ J such that p k + m k + l k − l J d k ≥ 0. Then m k + p k − l J d k ≥ −l k , so that H l k (P l k , O P l k (m k + p k − l J d k )) = 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 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 = l J . By the Künneth formula, it suffices to show that The first line of (1.4) is easy, since by Serre duality, For the second line of (1.4), suppose that k / ∈ J. By the maximality of J, we must have Note that l J∪{k} = l J + l k . If the maximum in (1.5) occurs at an element of J, say s ∈ J, then Hence the maximum occurs at k, so that It follows that the second line of (1.4) is nonzero, as desired.
If we fix the sheaf O X (m) and let L = O X (p) vary over all p ∈ Z r , we get the regularity set This set is easy to describe. Given a permutation σ in the symmetric group S r , let J(σ, k) ⊆ {1, . . . , r} denote the subset Then define We build a permutation σ ∈ S r as follows. Theorem 1.2 tells us that (1.1) holds for all J = ∅. Hence, for Setting σ(1) = k 1 , this becomes Setting σ(1) = k 2 , this becomes Continuing in this way gives σ ∈ S r with p ∈ p σ + N r .
This implies that l J ≤ l J(σ,k) . Since p ∈ p σ + N r , we obtain This proves that (1.1) holds for J, and since J = ∅ was arbitrary, we have p ∈ Reg(O X (m)) by Theorem 1.2.
We next apply our results to study the Castelnuovo-Mumford regularity of O X (m) under the projective embedding given by the Segre-Veronese map Hence the regularity of F (m) is given by for all J = ∅. From here, the first assertion of the theorem follows easily, and the second assertion is immediate.
We can derive this from Theorem 1.4 as follows. We know that I is λ-regular if and only if O Y is λ − 1 regular. Note also that λ can be defined more simply as If J = ∅, it follows easily that By Theorem 1.4, we obtain λ − 1 ≥ reg(F (0)) = reg(O Y ), so that λ ≥ reg(I), as claimed.
It is also easy to see that λ − 1 > reg(O Y ) can occur. For example, if r = 2 and d = (1, 1), then one can show without difficulty that

Subadditivity
In this section we study how the regularity of O X (m) and O X (m ′ ) compares to the regularity of the tensor product We first consider regularity as defined in Definition 1.1.
Proof. We will use Theorem 1.2. Given a nonempty subset J ⊆ {1, . . . , r}, it suffices to find k ∈ J such that (2.1) p k + p ′ k + m k + m ′ k + l k − l J d k ≥ 0. Using (1.1) for this J and the sheaves O X (m) and O X (p), we know that max s∈J p s + m s + l s − l J d s ≥ 0.
Hence we can find k ∈ J such that Then using (1.1) for {k} and the sheaves O X (m ′ ) and O X (p ′ ), we also have Proof. It suffices to show that if F (m) is p-regular and F (m ′ ) is p ′ -regular, then F (m + m ′ ) is (p + p ′ )-regular. This follows immediately from Theorem 2.1 and the already-noted equivalence In general, regularity is not subadditive, i.e., given coherent sheaves F and G on P N , the inequality may fail. Here is an example due to Chardin. As noted by Chardin [3], this remains true when we work in the larger ring S = k[x, y, z, t, u, v]. The key point is that the ideals I n , J m , and I n + J m are saturated in S. Now let F and G be the coherent sheaves associated to S/I n and S/J m respectively. Then F ⊗ G is the sheaf associated to S/(I n + J m ). Since I n and J m are saturated in S, (2.5) easily implies that reg(F ) + reg(G) = m + 2n − 3 < reg(F ⊗ G) = mn − 2 when n, m ≥ 3. This shows that subadditivity fails in general.
However, there are certain situations where (2.4) does hold, such as when F or G is locally free (see [7, Prop. 1.8.9]). Theorem 2.1 shows that the sheaves F (m) give another class of coherent sheaves for which regularity is subadditive.

Tate Resolutions
By [5], a coherent sheaf F on the projective space P(W ) has a Tate resolution · · · −→ T p (F ) −→ T p+1 (F ) −→ · · · of free graded E-modules, E = W * , with terms Here, E = Hom K (E, K) is the dual over the base field K. Standard vanishing theorems imply Furthermore, if F = i * E for a locally free sheaf E on an irreducible Cohen-Macaulay subvariety Y ֒→ P(W ), then Serre duality and the same vanishing theorems imply that T p (F ) = E(n − p) ⊗ H n (F (p − n)) for p ≪ 0.