OF CURVES IN GRASSMANNIANS
E. BALLICO
We study the Castelnuovo-Mumford regularity and the Grassmann regularity of any curveX embedded in a Grassmannian, mainly if the embedding comes from a stable and spanned vector bundle onX.
AMS 2000 Subject Classification: 14M15, 14H60, 14J60.
Key words: Grassmannian; Castelnuovo-Mumford regularity; Grassmann regula- rity; stable vector bundle.
1. INTRODUCTION
Chipalkatti [4] introduced a notion of regularity for coherent sheaves on a GrassmannianGand proved that his definition gives a nice resolution of any such sheaf ([4], Theorem 1.11), exactly as the Castelnuovo-Mumford regularity of a sheaf on a projective space is related to its minimal free resolution (a similar, but different, resolution is obtained by Costa and Mir´o-Roig ([5])) in a very general setting. In this paper we study the definition of regularity for coherent sheaves on the Grassmannians introduced in [4] and the usual Castelnuovo-Mumford regularity for curves X embedded in a Grassmannian G by a stable vector bundle. Fix integers n > k > 0 and an algebraically closed field Kwith char(K) = 0. Let G =G(k, n) denote the Grassmannian of all k dimensional linear subspaces of V := K⊕n. Let S (resp. Q) denote the tautological subbundle (resp. quotient bundle) on G. Hence rank(S) =k, rank(Q) =n−k, det(Q)∼= det(S∗)∼=OG(1) and there is an exact sequence
(1) 0→S →V ⊗ OG →Q→0
onG(the Euler sequence). LetAbe a coherent sheaf onG. Ais said to bem- regular in the sense of Castelnuovo-Mumford orm-CM-regular ifHi(G, A(m−
i)) = 0 for all i >0. For any finite non-increasing sequence λ= λ1 ≥ · · · of non-negative integers let Σλ(V) denote the Schur module associated toV ([4]).
Hence for any integer t≥0 and any vector bundleE onGwe have Σ(t)(E)∼= Symt(E) and Σ(1,...,1)(E)∼=Vt
(E), in which 1 is repeatedttimes in (1, . . . ,1).
A coherent sheafA onGis said to bem-regular in the sense of Chipalkatti or
REV. ROUMAINE MATH. PURES APPL.,54(2009),2, 117–123
m-GifHi(G, A(m)⊗Σβ(Q∗)) = 0 for all i >0 and all (n−k-ples of integers β := (β1, . . . , βn−k) such that k ≥ β1 ≥ · · · ≥ βn−k ≥ 0. The Castelnuovo- Mumford regularity (resp. G-regularity) is the minimal integer m such that A is m-regular (resp. m-G-regular), or −∞ if no such minimum exists. A has CM-regularity −∞ if and only if it has finite support, i.e., if and only if it has G-regularity −∞ [4, Remark 1.2.4]. The two regularities may be very different: the sheaf OG hasG-regularity 0 and CM-regularityk(n−k) + 1−n [4, Remark 1.2.2]. In Section 2 we will prove the following results.
Theorem1. Let F be a coherent sheaf onGsuch that its support T has dimension at most 1. IfF is m-CM-regular, then it is (m+k−1)-G-regular.
Corollary1. LetX ⊂Gbe an integral curve. Let mbe the dimension of the linear span ofXin the Pl¨ucker embedding ofG. ThenOX is(d−m+k)- G-regular, and IX,G is (d−m+k+ 1)-G-regular.
Roughly speaking, Corollary 1 is an improvement of [4, Theorem 3.2 with a term “d” instead of a term “2d”]. Sincedmay be very low even ifXis non- degenerate in the sense of [4], to see why Corollary 1 is nice we need to check that the integer mis not too low. Assume that X comes from an embedding hE :X →Gobtained in the following way. Fix a spanned rankrvector bundle E on X and set n:= h0(X, E) andk :=n−r. By the universal property of the Grassmannian, the pair (E, H0(X, E)) induces a morphismhE :X →G. Set d:= deg(E). In this case d= deg(E) and the integer m is computed as follows. Let α : V(n−k)
(V) → H0(X,det(E)) be the natural map associated with the Pl¨ucker embedding of G. Since m+ 1 = dim(Im(α) ([11], [2]), and in our set-up the integerh0(X,det(E)) =d+ 1−pa(X) is known, the best we may expect is the surjectivity of α. LetX be an integral projective curve. Set g:=pa(X) and assume g≥2 . For all integersr >0 and dletU(r, d) be the moduli space of all stable locally free sheaves on X with rank r and degreed;
U(r, d) is non-empty, smooth, integral, and dim(U(r, d)) =r2(g−1) + 1 (see [10, Remark on p. 167], for the case of a singular curve). In Section 2 we will prove the following result.
Proposition 1. Fix integers r≥2, andd.
(a) If d≥ rg+ 1 and E is general in U(r, d), then h1(X, E) = 0, E is spanned, hE is an embedding, and αE is surjective.
(b)If either d≥2rg+ 2r2−2r or d≥(2g−1)r and X is smooth, then h1(X, E) = 0, E is spanned, and hE is an embedding.
Remark 1. Fix (E, H0(X, E)) such thatE∈U(r, d),r ≥2,E is spanned andhE is an embedding. It is very easy to compute the Castelnuovo-Mumford regularity γ of hE(X) and the Castelnuovo-Mumford regularity δ of E seen as a coherent sheaf on Gsuch that E hashE as its support. The integerγ is the minimal integer t such that h1(X,det(E)⊗(t+1)) = 0. Hence γ ≥0 if and
only if g >0, γ = 1 if and only if det(E) is non-special, and d·γ ≤ 2g−2.
The integer δ is the minimal integer such thath1(X, E ⊗det(E)⊗(t+1)) = 0.
Riemann-Roch givesδ·d≥r(g−1). The stability of E givesδ·d < r(2g−2) (use Lemma 3 below and the fact that E⊗L is stable for allL∈Pic(X)).
2. PROOFS AND OTHER RESULTS
For any coherent sheaf A on G and any integer i ≥ 0 set Hi(A) :=
Hi(G, A) and hi(A) := dimKhi(A). Let η be the set of all partitions β :=
(β1, . . . , βn−k) such thatk≥β1≥ · · · ≥βn−k≥0. Set Φ :={Σβ(Q∗) :β∈η}.
Hence Φ are the test sheaves for the m-G-regularity: A ism-G-regular if and only if Hi(A(m)⊗D) = 0 for all D∈Φ.
Remark 2. Letj :G,→ PN be any embedding such thatj∗(OPN(1))∼= OG(1). For instance, we may take asj the Pl¨ucker embedding of G. LetAbe a coherent sheaf onG. Notice thatj∗(A(t))∼=j∗(A)(t) for everyt∈Z. Sincej is affine, the Leray spectral sequence ofjgivesHi(G, A(t))∼=Hi(PN, j∗(A(t))) for all (i, t)∈N×Z. HenceA ism-CM-regular as a sheaf onGif and only if j∗(A) ism-regular in the usual sense ([9, Chapter 14]).
Remark 3. Let A be a coherent m-CM-regular sheaf on G. A(m) is spanned and A is (m+ 1)-CM-regular (use Remark 5 and the classical case considered in [9, Chapter 14]. Hence Hi(G, A(t)) = 0 for all i > 0 and all t≥m−i.
Remark 4. Here we will check the well-known fact that Σβ(Q∗)(k) is spanned for every β := (β1, . . . , βn−k)∈η. Since Σβ(Q∗)(k) is a direct factor of Vβ1(Q∗)⊗ · · · ⊗ Vβn−k(Q∗(k), it is sufficient to prove that Vi(Q∗)(1) is spanned for all 0≤i≤n−k. This is true because the vector bundleVi(Q∗)(1) is associated with an irreducible representation with non-negative weights.
Proof of Theorem1. Fix anyβ ∈η. Since dim(T)≤1, we havehi(F(m−
1)⊗Σβ(Q∗)(k)) = 0 for alli≥2. Hence it is sufficient to proveh1(F(m−1)⊗
Σβ(Q∗)(k)) = 0. Since the coherentOG-sheaf F is supported byT, there is a closed subscheme A ⊂G such that F is a coherent OA-sheaf and Ared = T. Since Σβ(Q∗)(k) is locally free,F ⊗Σβ(Q∗)∼=F(m−1)⊗OA(Σβ(Q∗)(k)|A).
Since Σβ(Q∗)(k) is spanned, Σβ(Q∗)(k)|A is spanned. Hence there are an integerN >0 and a surjection ofOA-sheavesv:F(m−1)⊕N → F ⊗Σβ(Q∗).
Set M := Ker(v). Thus we have an exact sequence
(2) 0→M → F(m−1)⊕N → F ⊗Σβ(Q∗)→0
of OA-sheaves. Since M is a coherent OA-sheaf and dim(A) ≤ 1, we have h2(A, M) = 0. SinceF ism-CM-regular, we haveh1(A, F(m−1)) = 0. Hence (2) gives h1(F(m−1)⊗Σβ(Q∗)(k)) = 0.
Proof of Corollary 1. Let J be the ideal sheaf of X in the Pl¨ucker embedding j : G → P := P(Vn−k
(V)) of G. By [6, Theorem 1.1], J is (d−m + 2)-regular in the sense of Castelnuovo-Mumford. Since OP is 0- regular in the sense of Castelnuovo-Mumford, a standard exact sequence gives that OX is (d−m+ 1)-regular in the sense of Castelnuovo-Mumford. Apply Theorem 1.
Remark 5. Fix a generalE ∈U(r, d). It is well-known thath0(X, E) = 0 and h1(X, E) =r(g−1)−difd≤r(g−1) while h0(X, E) =d+r(1−g) and h1(X, E) = 0 ifd≥r(g−1).
From now on we will always assume r ≥ 2. For any rank r vector bundle E on X and any linear subspace V ⊆ H0(X, E) let αE,V : Vr
V → H0(X,det(E)) be the natural map. Set αE :=αE,H0(X,E).
Lemma1. Fix integersr≥2andai ≥g+1,1≤i≤r. LetLi,1≤i≤r, be a general element of Picai(Li). Then h0(X, Li) =ai+ 1−g, h1(X, Li) = 0, Li is spanned, and the multiplication map µL1,...,Lr : H0(X;L1) ⊗ · · · ⊗ H0(X, Lr)→H0(X, L1⊗ · · · ⊗Lr) is surjective.
Proof. Everything is obvious and well-known, except the surjectivity of µL1,...,Lr. To prove the surjectivity of µL1,...,Lr we first assume r = 2. Notice thath1(X, L1⊗L2) = 0, henceh0(X, L1⊗L2) =a1+a2+ 1−g=h0(X, L1) + h0(X, L2) + 1−g. First, assumea1 =g+ 1. Hence h0(X, L1) = 2. Since L1
is spanned, we obtain the exact sequence
(3) 0→L2⊗L∗1 →L⊕22 →L1⊗L2 →0.
FixL2 and then take a general degreeg+ 1 line bundleL1. We geth0(X, L2⊗ L1) = a2 −(g+ 1) + (1−g). Hence the cohomology exact sequence of (3) gives rank(µL1,L2) = 2(a2+ 1−g)−a2−2g =a2+a1+ 1−g. Hence µL1,L2
is surjective in this case. Now, assume a1 > g+ 1. We use induction on a1. Fix a general P ∈ X. Hence OX(P) is locally free and L1(−P) is general in Pica1−1(L1). By the inductive assumption, µL1(−P),L2 is surjective. Hence rank(µL1(−P),L2) =a1+a2−g. The multiplication by P induces an inclusion H0(X, L1(−P)),→ H0(X, L1) and H0(X, L1⊗L2(−P)) ,→H0(X, L1⊗L2).
Using these inclusions, we see that Im(µL1(−P),L2) =H0(X, L1⊗L2(−P)), as a codimension one linear subspace of H0(X, L1⊗L2) contained in the set of sections vanishing at P. Since P is not a base point of L1 orL2, Im(µL1,L2) strictly contains this hyperplane. Hence µL1,L2 is surjective. Now, assume r >2. The line bundleM :=L2⊗ · · · ⊗Lr is spanned and non-special. Apply the first part of the proof to the line bundles L1 andM to get the surjectivity of µL1,M and then use the fact that Im(µL2,...,Lr) =H0(X, M).
Remark 6. Fix integersr ≥2 andai ≥g+1, 1≤i≤r. LetLi, 1≤i≤r, be a general element of Picai(Li). SetF :=⊕ri=1Li. Henceh1(X, F) = 0. The surjectivity of the multiplication mapµL1,...,Lr :H0(X;L1)⊗· · ·⊗H0(X, Lr)→ H0(X, L1⊗ · · · ⊗Lr) (Lemma 1) gives the surjectivity ofαF.
Remark 7. LetE be a rankr vector bundle onX. By [7, Corollary 2.2], E is a flat limit of a family of stable vector bundles on X, at least if X is smooth. For reader’s sake we copy the proof in [7]. Since there is nothing to prove if r = 1, we assume r ≥2. Fix P ∈Xreg and any rank r stable vector bundle F on X such that det(F) ∼= det(E). Fix a large integer k such that E(kP) and F(kP) are spanned. Taking a general (r−1)-dimensional linear subspace of H0(X, E(kP)) and of H0(X, F(kP)), we see that bothE and F fit as middle terms in an extension of det(E(kP)) by OX(−kP)⊕(r−1). Since the space of all such extensions is a vector space V, it is an integral variety.
Since on the integral varietyVthere is a tautological family of extensions (see [8]), it is sufficient to use the fact that stability is an open condition.
Lemma 2. Fix integers r ≥ 2 and d ≥ r(g+ 1). Let E be a general element of U(r, d). Then αE is surjective.
Proof. Seta1:=d−(r−1)(g+ 1) andai :=g+ 1 for 2≤i≤r. TakeF as in Remark 6. Since F is a flat limit of a flat family of stable vector bundles (Remark 7) and the surjectivity of αEλ is an open property for flat families {Eλ} of non-special vector bundles, it is sufficient to use Remark 2.
Lemma3. Fix integers r≥2 andd≥r(2g−2). Then h1(X, E) = 0 for all E∈U(r, d).
Proof. Fix E ∈ U(r, d) and assume h1(X, E) > 0, i.e., assume the existence of a non-zero morphism u : E → ωX (use the fact that X is lo- cally Cohen-Macaulay and duality (see [1])). Duality and Riemann-Roch give deg(ωX) = 2g−2. Hence the existence ofu6= 0 contradicts the stability ofE and the assumption µ(E) =d/r ≥2g−2.
Lemma 4. Fix integers r, d such that r ≥2 and d≥(2g−1)r. Assume that X is smooth and fix any E ∈ U(r, d). Then h1(X, E) = 0, h0(X, E) = d+r−rg, E is spanned, and the morphism hE :X → G(d−rg, d+r−rg) induced by the pair (E, H0(X, E))is an embedding.
Proof. Since h1(X, E) = 0 (Lemma 3), Riemann-Roch givesh0(X, E) = d+r−rg. Fix any P ∈X. Since X is smooth, IP ⊗E ∈U(r, d−r). Hence h1(X,IP ⊗E) = 0 (Lemma 3). Hence a standard exact sequence gives thatE is spanned at P. Since E is spanned, to check thathE is an embedding it is sufficient to check h0(X,IZ⊗E) ≤h0(X, E)−r−1 for any length 2 closed subschemeZ ⊂X. Fix any suchZ and takeP ∈Zred. Sinceh0(X,IP⊗E) =
h0(X, L)−r, it is sufficient to prove h0(X,IZ⊗E)< h0(X,IP⊗E). Assume h0(X,IZ ⊗E) = h0(X,IP ⊗E), i.e. assume h1(X,IZ ⊗E) = r (Riemann- Roch). Set F :=ωX⊗(IZ⊗E)∗. F is a rankr stable vector bundle such that deg(F) =r(2g−2)−d+ 2r = 2rg−d≤r. Serre duality givesh0(X, F) =r, contradicting [3], Theorem B.
In the singular case we only know the following weaker statement.
Lemma 5. Fix integers r≥2 and d≥r(2g−2), and any E∈U(r, d).
(a)If d≥2rg+r2−2r, then E is spanned.
(b) If d ≥ 2rg+ 2r2 −2r, then E is spanned, and the morphism hE : X →G(d−rg, d+r−rg)induced by the pair(E, H0(X, E)) is an embedding.
Proof. This is a trivial modification of the proof of [10, Lemma 5.2, page 166]. Lemma 3 gives h1(X, E) = 0. Fix any P ∈X. To prove part (a), it is sufficient to prove that E is spanned at P. Since E is locally free, the defining sequence of OP induces the exact sequence
(4) 0→ IP ⊗E →E →E|{P} →0.
Since E has rank r and it is locally free, h0(X, E|{P}) =r. Hence (4) gives that the torsion-free sheaf IP ⊗E has degree d−r. Since h1(X, E) = 0, E is spanned at P if and only if h1(X,IP ⊗E) = 0. Assume h1(X,IP ⊗E)>
0. Hence by duality there is a non-zero map v : IP ⊗E → ωX ([1]). Set M := Ker(u). Since deg(ωX) = 2g−2, deg(M) ≤ d−r −2g+ 2. M is a rank r −1 subsheaf of E. Since E is stable, we get µ(M) < µ(E), i.e.
rd−r2−2rg+ 2r < rd−d, i.e. d < r2−2r+ 2rg, contradiction. To check part (b) it is sufficient to prove h1(X,IZ⊗E) = 0 for every length 2 closed subscheme Z of X. We mimic the proof of part (a) takingIZ⊗E instead of IP ⊗E. Since IZ⊗E is a degreed−2r subsheaf of E the new kernel M is a rank r−1 subsheaf ofE with degree at least d−2r+ 2g−2. The stability of E gives a contradiction ifd≥2rg+ 2r2−2r.
Proof of Proposition1. Part (a) follows from Lemmas 3 and 2. Part (b) follows from Lemmas 5 and 4.
Acknowledgements. The author was partially supported by MIUR and GNSAGA of INdAM (Italy).
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Received 26 May 2008 University of Trento
Department of Mathematics 38050 Povo (TN), Italy
