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Stability of sheaves over Quot schemes
Accepted Manuscript Stability of sheaves over Quot schemes
Chandranandan Gangopadhyay
PII: DOI: Reference:
S0007-4497(18)30075-7 https://doi.org/10.1016/j.bulsci.2018.08.001 BULSCI 2761
To appear in:
Bulletin des Sciences Mathématiques
Received date:
1 June 2018
Please cite this article in press as: C. Gangopadhyay, Stability of sheaves over Quot schemes, Bull. Sci. math. (2018), https://doi.org/10.1016/j.bulsci.2018.08.001
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STABILITY OF SHEAVES OVER QUOT SCHEMES CHANDRANANDAN GANGOPADHYAY
Abstract. Let C be a smooth projective curve over an algebraically closed field k. Let r Q(r, d) be the Quot scheme of degree d zero dimensional quotients of OC . Let A(r, d) be the kernel of the universal quotient map over C × Q(r, d) and let πQ be the second projection from C × Q(r, d) to Q(r, d). Then, we prove that with respect to certain natural polarisations on C × Q(r, d) and Q(r, d), A(r, d) and (πQ )∗ A(r, d) are stable. A r is also proved. similar statement over Flag scheme of torsion quotients of OC
1. INTRODUCTION Fix r, d ≥ 1. Let C be a smooth, projective curve over an algebraically closed field k. Let Q(r, d) be the Quot scheme of degree d zero dimensional quotients of OCr . Over C × Q(r, d), we have the universal exact sequence: 0
A(r, d)
B(r, d)
r OC×Q(r,d)
0
Let πC and πQ are the projections from C × Q to C and Q respectively. Then, if P ∈ C, then OC (P ) is ample, and det((πQ )∗ (B(r, d) πC∗ OC (lP ))) is ample on Q(r, d) for l >> 0. This follows from [1, Chapter 2, Proposition 2.2.5]. Then, we prove that: Theorem 4.1. With respect to the above polarisation, the coherent sheaf (πQ )∗ A(r, d) is stable. Also, for a, b > 0, the line bundle (πC∗ OC (P ))a is ample on C × Q(r, d). Then, we prove that
∗ (det ((πQ )∗ (B(r, d) πQ
πC∗ OC (lP )))b )
Theorem 3.3. (i) Suppose d ≥ 2. Then, if da + (d + r − rd)bl ≥ 0, then with respect to the above ample line bundle A(r, d) is semistable, and if the inequality is strict, it is stable. (ii) If d = 1, then if ral − bl − a ≥ 0, then A(r, d) is semistable. If the inequality is strict, then it is stable. Infact, the analogous statement to Theorem 4.1. in the case of Flag schemes of filtrations of torsions quotients of OCr is also true [Theorem 5.1]. The proof is essentially the same as in the case of Quot schemes. We give an outline of the proof in the fifth section. 1
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CHANDRANANDAN GANGOPADHYAY
Note that if r = 1, then, A(r, d) ⊆ OC×Q(r,d) and (πQ )∗ A(r, d) ⊆ (πQ )∗ OC×Q(r,d) = OQ(r,d) , and hence, they have rank 1 or 0. Therefore, they are stable. So, from here onwards, we assume that r ≥ 2. We will be using the following lemma [2, Lemma 1.1] to reduce the problem to checking semistability(or stability) of certain sheaves over C × (C × Pr−1 )d and (C × Pr−1 )d : Lemma 1.1. [2, Lemma 1.1] Let X and Y are two smooth projective varieties of same dimension. Let f : X Y be a dominant rational map defined outside a subset Z ∈ X of codim ≥ 2. Let DX and DY be ample divisors on X and Y respectively such that f ∗ DY |X\Z = DX |X\Z . Let E be a vector bundle on Y such that f ∗ E extends to a vector bundle F on X. Then, if F is DX -semi-stable(resp. stable), then E is DY -semi-stable (resp. stable). To use the above lemma, we first show that there exists a rational map Φ : (C × Pr−1 )d Q(r, d), which is defined outside the closed subset of codimension 2.
2. Construction of the map Φ Notation: Let X = (C × Pr−1 )d , let the i-th projection to Pr−1 be denoted by pi,Pr−1 and the i-th projection to C be denoted by pi,C . For {i1 , i2 , ..., ik } ⊆ {1, 2, ..., d}, k ≥ 2, let i1 ,i2 ,...,ik = {x ∈ X |pi1 ,C (x) = pi2 ,C (x) = ... = pik ,C (x), pi1 ,Pr−1 (x), pi2 ,Pr−1 (x).., pik ,Pr−1 (x) are linearly dependant}. Let Z = i1 ,i2 ,...,ik and let U = X \Z. Note that each of these i1 ,i2 ,..,ik has codimension ≥ 2 and hence, Z has codimension 2 in X . Let π1 and π2 be the first and second projection from C × X to C and X respectively, and let pi,C ◦ π2 be denoted by πi,C and pi,Pr−1 ◦ π2 be denoted by πi,Pr−1 . Now, over C × X , let q be the following map:
q:
r OC×X
→
d
∗ πi,P r−1 O(1)|0,i,C
(2.1)
i=1 r ∗ where 0,i,C = {(c, x) ∈ C × X |c = πi,C (x)} and the quotient OC×X → πi,P r−1 O(1)|0,i,C is r−1 r , O → O(1) by πi,Pr−1 followed obtained by pulling back the natural quotient over P Pr−1 by the restriction to 0,i,C .
Theorem 2.1. There exists a dominant map Φ : U → Q(r, d) such that: r (i) (idC × Φ)∗ (OC×Q(r,d) → B(r, d)) = q|C×U ∗ ∼ (ii) (idC × Φ) A(r, d) = ker q|C×U d (iii) Φ∗ det ((πQ )∗ (B(r, d) πC∗ OC (lP )))= (p∗i,C OC (lP ) p∗i,Pr−1 OPr−1 (1))|U . i=1
STABILITY OF SHEAVES OVER QUOT SCHEMES
3
Proof. (i) By the universal property of Quot schemes, it is enough to show that q|C×U is ∗ surjective, and di=1 πi,P r−1 O(1)|0,i,C , when restricted to C × {x}, for x ∈ X , is a torsion sheaf of degree d. d ∗ If x = ((c1 , v1 ), (c2 , v2 ), .., cd , vd ) ∈ X , then πi,P r−1 O(1)|1,i,C when restricted to C × {x} is nothing but
d i=1
i=1
kci , hence it is of degree d.
Claim: q|C×U is a surjection.
r Proof. Now let x = ((ci , vi )) ∈ U . Then, over C×{x}, q|C×{x} is given by OC×{x} →
d i=1
k ci ,
where, each OCr → kci is given by vi , i.e. if we choose homogeneous co-ordinate of vi = (0) (1) (r−1) (0) (1) [vi , vi , .., vi ], then, the map OCr → kci is given by: e(0) → vi , e(1) → vi , .., e(r−1) → (r−1) . vi We only need to check surjectivity at the points c1 , c2 , .., cd . Without loss of generality, assume c1 = c2 = .. = cj = c and c = ci ∀i ≥ j + 1. Then, q|{c}×{x} is given by the map: k r → k j , with (0) (0) (0) (1) (1) (1) (r−1) (r−1) (r−1) , v2 , .., vj ), i.e. e0 → (v1 , v2 , .., vj ), e1 → ((v1 , v2 , .., vj )),..., er−1 → (v1 ⎡ ⎤ (0) (1) (r−1) v v .. v1 ⎢ 1(0) 1(1) (r−1) ⎥ ⎢v ⎥ v2 .. v2 the matrix of this map is given by: ⎢ 2 ⎥ Now, since x ∈ U , hence, the . .. .. .. ⎣ ⎦ (0) (1) (r−1) vj vj .. vj rows are linearly independant. Therefore, rank of the above matrix is j. Hence, the above map is surjective at c. To see that Φ is dominant, first note that if x ∈ Q(r, d) corresponds to a quotient d kci with all ci ’s distinct, then, Φ−1 (x) is a non-empty finite set. Therefore, OCr → i=1
dim Image Φ = dim X = rd. Now, since, Q(r, d) is a smooth projective variety of dimension rd, so, Image Φ = Q(r, d). (ii) Note that there is a surjection from (idC × Φ)∗ A(r, d) → (ker q)|C×U . So, the claim follows from the fact that both of these sheaves are vector bundles of same rank. This fact is proved in Lemma 1.1. (iii) This is proved in two steps: ∗ Step 1: Φ (πQ )∗ (B(r, d) πC∗ O(lP )) ∼ = (π2 )∗ (idC × Φ)∗ (B(r, d) πC∗ O(lP )). d Step 2: (π2 )∗ (idC × Φ)∗ (B(r, d) πC∗ O(lP )) ∼ = (p∗i,C OC (lP ) p∗i,Pr−1 OPr−1 (1))|U . i=1
Then, (iii) will follow by taking the top exterior powers of the above bundles. Proof of Step 1: Let G = B(r, d) πC∗ O(lP ). We have the natural map Φ∗ (πQ )∗ (G) →
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CHANDRANANDAN GANGOPADHYAY
(π2 )∗ (idC × Φ)∗ (G) such that if u ∈ U , then we have the following commutative diagram: (Φ∗ (πQ )∗ G)|u = ((πQ )∗ G)|Φ(u)
((π2 )∗ (idC × Φ)∗ G)|u H 0 (C, G|C×{u} )
Since, for all x ∈ Q(r, d), G|C×{x} is a torsion sheaf of degree d, hence, by [3, Chapter III, Corollary 12.9], the map ((πQ )∗ G)|x → H 0 (C, G|C×{u} ) is an isomorphism. Similarly,the vertical arrow in the above diagram is also an isomorphism. Hence (Φ∗ (πQ )∗ G)|u → ((π2 )∗ (idC × Φ)∗ G)|u is an isomorphism. Therefore (Φ∗ (πQ )∗ G) → ((π2 )∗ (idC × Φ)∗ G) is an isomorphism. Proof of Step 2: d d ∗ ∗ ∗ ∗ π1 OC (lP )) = (π2 )∗ (πi,P π1 OC (lP )). (π2 )∗ ( πi,P r−1 O(1)|0,i,C r−1 O(1)|0,i,C =
d i=1
i=1
∗ (π2 )∗ (πi,P r−1 O(1)|0,i,C
i=1
π1∗ OC (lP )|0,i,C ).
Now, 0,i,C = X and if j :0,i,C → C × X is the inclusion, then π2 ◦ j is identity, π1 ◦ j = pi,C . Hence, πi,Pr−1 ◦ j = pi,Pr−1 and ∗ ∗ ∗ ∗ O(1)| j π1 OC (lP )) π1∗ OC (lP )|0,i,C ) = (π2 )∗ (j∗ )(j ∗ πi,P (π2 )∗ (πi,P r−1 r−1 O(1) 0,i,C ∗ ∗ = pi,Pr−1 O(1) pi,C OC (lP ).
Lemma 2.2. Let Y be a variety over k. Over C ×Y , suppose we are given a surjection; r → F such that for any y ∈ Y , F|C×{y} is a torsion sheaf over C of degree d. f : OC×Y r is an injection. Then, ker f is a vector bundle of rank r and (ker f )|C×{y} → OC×{y} Proof. Since, for all y ∈ Y , the Hilbert polynomial of F|C×{y} with respect to any polarir is also sation is the constant polynomial d, it follows that F is flat over Y . Since, OC×Y flat over Y so, ker f is also flat over Y . Now, flatness of ker f implies for all y ∈ Y , (ker f )|C×{y} ∼ = ker (f |C×{y} ). This alr ready implies the injectivity of ker f )|C×{y} → OC×{y} . Also this sheaf, being a subsheaf of OCr , is locally free, and since its cokernel is torsion, it has rank r. Hence, for all (c, y) ∈ C × Y , (ker f )|{c}×{y} is a vector space of rank r. Since, the rank remains constant for all (c, y) ∈ C × Y , this implies that ker f is a vector bundle.
Notation: Let us denote ker q by F(r, d). Corollary 2.3. If F(r, d) is semistable with respect to the polarisation d ∗ ∗ π1∗ OC (a) (πi,C OC (lb) πi,P r−1 OPr−1 (b)), then A(r, d) is semistable with respect to i=1 ∗ (det ((πQ )∗ (B(r, d) πC∗ OC (lP )))b ). (πC∗ OC (P ))a πQ
STABILITY OF SHEAVES OVER QUOT SCHEMES
5
[For notational convenience, we denote OC (lP ) by OC (l).]
Proof. Since, Q(r, d) is a smooth projective variety of dimension rd = dim X , so this corollary follows from Lemma 1.1 and Theorem 2.1. 3. Semistability and Stability of A(r, d) From now on, we will denote by (a, b, c) the polarisation on C × X corresponding to the d ∗ ∗ πi,C O(b) πi,P line bundle π1∗ O(a) r−1 O(c). First, we will determine a condition on i=1
(a, b, c) such that F(r, d) is semistable(or stable ) with respect to (a, b, c)[Theorem 3.2.4] and then use Corollary 2.3 to find polarisations on C × Q(r, d) with respect to which A(r, d) is semistable(or stable). 3.1 Degree of Sheaves with respect to the polarisation (a,b,c)
Theorem 3.1.1. Let X1 , X2 , ..., Xn are smooth projective varieties of dimension d1 , d2 , ..., dn respectively. Let DXi be an ample divisor on Xi . If pi : X1 ×X2 ×...×Xn → Xi n is the i-th projection to Xi , then p∗i DXi is an ample divisor of X1 × X2 × .. × Xn . Let i=1
L be a line bundle on X1 × X2 × .. × Xn . Then,with respect to deg L =
n
i=1
n
(d1 + d2 + .. + dn − 1)! deg d1 !d2 !..(di − 1)!dn! j=1,j=i
n i=1
DXj (DXj )
p∗i DXi , deg
DXi L|Si ,
(3.1)
where, if for each i, if we have any choice of xi,j ∈ Xj , j = i, then Si = {x ∈ X1 × X2 × ... × Xn |pj (x) = xi,j , ∀j = i}. Proof. Let Y is a smooth projective variety of dimension m, We denote the chow group of codimension i-cycles on Y by CH i (Y ). Since, Y is smooth, we have the intersection product : CH i (Y ) × CH j (Y ) → CH i+j (Y ). Also, since Y is projective, we have the degree map deg : CH n (Y ) → Z, given by k k [ ni Pi ] → ni , where Pi ∈ Y . i=1
i=1
Let L be a line bundle on Y and let D be an ample divisor on Y . Let us denote the class of L and D in CH 1 (Y ) by [L] and D respectively. Then, degree of L with respect to the ample divisor D is given by : deg([L].[D]m−1 ). Hence, in the context of our theorem, deg L with respect to degree of the following cycle :
n i=1
p∗i DXi , is given by the
6
[L].[
CHANDRANANDAN GANGOPADHYAY n i=1
[
p∗i DXi ](d1 +d2 +..+dn −1) ∈ CH d1 +d2 +..+dn (X1 × X2 × ... × Xn ). Now,
n
p∗i DXi ](d1 +d2 +..+dn −1)
=[
n
(i1 +i2 +..in ) =d1 +d2 +..+dn−1
i=1
=[
(d1 + d2 + ..dn − 1)! ∗ ij pj (DXj )] i1 !i2 !..in ! j=1 n
(i1 +i2 +..in ) =d1 +d2 +..+dn−1
=[
(d1 + d2 + ..dn − 1)! ∗ (pj DXj )ij ] i1 !i2 !..in ! j=1
d n
(d1 + d2 + ..dn − 1)! i=1
d1 !d2 !..(di − 1)!..dn !
(3.2)
di −1 p∗j (DXjj )p∗i (DX )] i d
j=1 j=i
The third equality follows from the fact that, if for some j, ij ≥ dj + 1, then (DXj )ij ∈ CH ij (Xj ) = 0. Hence, it is enough to show that the n n d di −1 degree of [L].[ p∗j (DXjj )p∗i (DX )] = deg i j=1,j=i
j=1 j=i
DXi (DXi )
deg
DXi L|Si .
d
Without loss of generality, assume i = 1. Then, for each 2 ≤ j ≤ n, let [DXjj ] = where xj,l ∈ Xj . Note that, by definition, deg Then, [
n
d1 −1 p∗j (DXjj )p∗1 (DX )] = [ 1 d
j=2
n
n j=2
p∗j (
kj l=1
DXj (DXj )
= kj .
d1 −1 p∗j (xj,l ))p∗1 (DX )] = 1
kj
[xj,l ],
l=1
n d1 −1 [ p∗j (xj,lj )p∗1 (DX )]. 1 lj j=2
d1 −1 p∗j (xj,lj )p∗1 (DX )] = [[(DX1 )d1 −1 ] × {x2,l2 } × {x3,l3 } × .. × {xn,ln }]. This follows 1 j=2 from the fact that if Y and Z are two smooth projective varieties, then CH ∗ (Y ) CH ∗ (Z) → CH ∗ (Y × Z) is a ring homomorphism. Now, by projection formula, [L][[(DX1 )d1 −1 ]×{x2,l2 }×{x3,l3 }×..×{xn,ln }] = [L|X1 ×{x2,l2 }×{x3,l3 }×..×{xn,ln } ].[(DX1 )d1 −1 ] = deg DX1 L|S1 . n d d1 −1 )] = deg DX1 L|S1 = k2 k3 ..kn (deg DX1 L|S1 ). Since, kj = Hence, [L][ p∗j (DXjj )p∗1 (DX 1
Now, [
j=2
deg
DXj (DXj ),
lj
we are done.
Corollary 3.1.2. Under the hypothesis of the above theorem, if F is a coherent sheaf on X1 × X2 × .. × Xn , then: deg F = deg F =
n n
(d1 + d2 + .. + dn − 1)! i=1
d1 !d2 !..(di − 1)!dn!
j=1,j=i
deg
DXi (DXi )
deg
DXi F |Si ,
(3.3)
STABILITY OF SHEAVES OVER QUOT SCHEMES
7
where, for each i, we have a generic choice of xi,j ∈ Xj , j = i, and Si = {x ∈ X1 × X2 × ... × Xn |pj (x) = xi,j , ∀j = i}.
Proof. Since X1 × X2 × .. × Xn is projective, F has a locally free resolution: 0 Fk Fk−1 ... F0 F 0 where, Fi are locally free. k l Then, det F = (det Fl )(−1) and deg F = deg (det F ). l=1
ˆi × .. × Xn such that Now, for each i, there exists a non-empty open set Ui ⊆ X1 × X2 × ..X n F |Ui ×Xi is flat over Ui . Hence, for any point (xi,j )j=1 ∈ Ui , the above resolution restricted j=i
to Si , remains exact, and gives a resolution of F |Si . In other words, det (F |Si )= (det F )|Si . Now, applying the above theorem for the line bundle det F , we get the required result. Corollary 3.1.3. Let G is a coherent sheaf over C × X . Then,
(rd)!bd−1 (cr−1 )d−1 cr−2 [bc deg G| + ac deg G|Si,C deg G = S 0,C ((r − 1)!)d i=1 d
+
d
(3.4)
(ab)(r − 1) deg G|Si,Pr−1 ]
i=1
Here, by deg G|Si,Pr−1 , we mean degree with respect to OPr−1 (1).
Proof. Using the above corollary, the degree of G with respect to the polarisation (a,b,c) is given by :
(rd)! (rd)! d r−1 d b (c ) deg G| + abd−1 (cr−1 )d deg G|Si,C deg G = S 0,C d ((r − 1)!)d ((r − 1)!) i=1 d
d
(rd)! abd (cr−1 )d−1 degc G|Si,Pr−1 d−1 (r − 2)! ((r − 1)!) i=1 (3.5) By degc G|Si,Pr−1 we mean degree of G|Si,Pr−1 with respect to O(c). Let us denote deg1 G|Si,Pr−1 by deg G|Si,Pr−1 . Then, degc G|Si,Pr−1 = cr−2 deg G|Si,Pr−1 . Then, we have +
(rd)!bd−1 (cr−1 )d−1 cr−2 [bc deg G| + ac deg G|Si,C S0,C ((r − 1)!)d i=1 d
deg G =
+
d
i=1
(ab)(r − 1) deg G|Si,Pr−1 ]
(3.6)
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CHANDRANANDAN GANGOPADHYAY
Notation: Let us denote the constant denote the
(rd)!bd−1 (cr−1 )d−1 cr−2 by K. Then, we will ((r − 1)!)d
μ(G) deg G by deg G and by μ (G). K K
Corollary 3.1.4. deg F(r, d) = −d(ac + bc)
Proof. Since the following sequence is exact outside a codimension 2 subset, we can compute deg F(r, d):
F(r, d)
0
i.e. deg F(r, d) = −
d i=1
d
q
r OC×X
i=1
∗ πi,P r−1 O(1)|0,i,C
∗ deg πi,P r−1 O(1)|0,i,C = −d(ac + bc).
(3.7)
3.2 Semistability and Stability of F(r, d)
d
r ∗ → πi,P Over C × X , let us denote ker (OC×X r−1 O(1)|0,i,C ) by Gi . Clearly, F(r, d) =
i=1
Gi .
Now, we have the following commutative diagram:
0 I0,i,C (1) 0
∗ πi,P r−1 Ω(1)
r OC×X
∗ πi,P r−1 O(1)
0
0
Gi
r OC×X
∗ πi,P r−1 O(1)|0,i,C
0
0
STABILITY OF SHEAVES OVER QUOT SCHEMES
9
Using snake lemma we have the following exact sequence: 0
∗ πi,P r−1 Ω(1)
Gi
I0,i,C (1)
0
(3.8)
Now, let H ⊆ F(r, d) be a subsheaf such that F(r, d)/H is torsion free. Then we have the following lemma: Lemma 3.2.1. For generic choices of c, ci ∈ C , vi ∈ Pr−1 , we have (i) deg H|Si,C ≤ 0, deg H|Si,Pr−1 ≤ 0 ∀i ∈ {1, 2, .., d} ∗ If H ⊆ πj,P r−1 Ω(1), then (ii)deg H|Sj,Pr−1 ≤ −1 ∗ If H πj,P r−1 Ω(1), then (iii)deg H|Sj,C ≤ −1, deg H|S0,C ≤ −1. Proof. First, we claim that for generic choices of c, ci ∈ C, vi ∈ Pr−1 , (a)the maps H|Si,C → F(r, d)|Si,C , H|Si,Pr−1 → F(r, d)|Si,Pr−1 , H|S0,C → F(r, d)|S0,C are all injective outside a closed subset of codimension ≥ 2. (b)the maps F(r, d)|Si,C → OCr , F(r, d)|Si,Pr−1 → OPr r−1 , F(r, d)|S0,C → OCr are all injective. Claim(a) will follow from Lemma 3.2.2. Proof of Claim (b). First we prove that for a generic choice of c, cj ∈ C and vi ∈ Pr−1 , F(r, d)|Si,C → OCr is injective. Without loss of generality, assume i = 1. Consider the map C × (C × Pr−1 )d
given by π1 × π1,Pr−1
C × Pr−1 × (C × Pr−1 )d−1 × (π2,C × π2,Pr−1 ) × ... × (πd,C × πd,Pr−1 ).
Consider the open set V ⊆ C × Pr−1 × (C × Pr−1 )d−1 of points of the form (c, v1 , (c2 , v2 ), .., (cd , vd )) with no two of c, ci equal. d r ∗ Then, when we restrict q : OC×X → i=1 πi,Pr−1 O(1)|0,i,C to the fibre of a point (c, v1 , (c2 , v2 ), .., (cd , vd )) ∈ V , it becomes OCr → kc given by the element v1 . Then, by Lemma 2.2, we are done. Similarly, we can show the injectivity of the other morphisms.
Proof of Lemma 3.2.1(contd.) (i) Combinig Claim (a) and (b), we get H|Si,C ⊆ OCr . So, deg H|Si,C ≤ 0. Similarly, deg
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CHANDRANANDAN GANGOPADHYAY
H|Si,Pr−1 ≤ 0. ∗ (ii) Without loss of generality, assume j = 1. Then, H ⊆ π1,P r−1 Ω(1). Now, by claim (a), the compostion ∗ H|S1,Pr−1 → (F(r, d) ∩ π1,P → F(r, d) is injective outside a closed subset of r−1 Ω(1))|S 1,Pr−1 ∗ is injective outside a codimension ≥ 2. Hence, H|S1,Pr−1 → (F(r, d) ∩ π1,P r−1 Ω(1))|S 1,Pr−1 closed subset of codimension ≥ 2. Now, consider the map C × (C × Pr−1 )d C × C × (C × Pr−1 )d−1 given by π1 × π1,C × (π2,C × π2,Pr−1 ) × ...(πd,C × πd,Pr−1 ). Let V ⊆ C×C×(C×Pr−1 )d−1 is the open set with points of the form (c, c1 , (c2 , v2 ), ..(cd , vd )) with all c, ci ’s distinct. Then, over the inverse image of V with respect to the above map, ∗ i.e. the open set Pr−1 ×V , F(r, d)|Pr−1 ×V = OPr r−1 ×V . Hence, (F(r, d)∩πi,P r−1 Ω(1))|Pr−1 ×V = ∗ ∗ π1,Pr−1 Ω(1)|Pr−1 ×V . Therefore, with respect to points in V , F(r, d) ∩ π1,Pr−1 Ω(1))|S1,Pr−1 ∼ = Ω(1). Therefore, H|S1,Pr−1 → Ω(1) is injective outside a codimension 2 subset, and hence, deg H|S1,Pr−1 ≤ −1. ∗ (iii) Without loss of generality, we can assume that j = d. Then, since H πd,P r−1 Ω(1), hence, using the exact sequence (2.1) we have a non-zero map H → I0,d,C (1). Now, consider the map : C × (C × Pr−1 )d C × (C × Pr−1 )d−1 × Pr−1 given by π1 × (π1,C × π1,Pr−1 ) × ...(πd−1,C × πd−1,Pr−1 ) × πd,Pr−1 . By Lemma 3.2.3, for a generic choice of (c, (c1 , v1 ), .., (cd−1 , vd−1 ), vd ), the above non-zerp map H → I0,d,C (1) when restricted to Sd,C is non-zero, i.e. we have a non-zero map H|d,C → Ic , where the map is given by the composition H|d,C → OCr → OC , where the map OCr → OC is given by vd . Also, combining this with claim (a) and (b), we also get that H|Sd,C → OCr is injective, in particular, the kernel of the map H|d,C → Ic is also contained in Or . Hence, deg H|d,C ≤ deg Ic = −1. Similarly, we can show that deg H|0,C ≤ −1. Lemma 3.2.2. Let Y and Z are two smooth varieties over k. Let F be a torsion-free coherent sheaf on Y × Z and let G ⊆ F be a subsheaf such that F/V is also torsion free. Then, for a generic y ∈ Y , G|{y}×Z → F |{y}×Z is injective outside a closed subset ⊆ {y} × Z of codimension ≥ 2.
STABILITY OF SHEAVES OVER QUOT SCHEMES
11
Proof. Let W ⊆ Y × Z be the closed subset outside whic F, G and F/G are locally free. Let U = (Y ×Z)\W . Since Y ×Z is smooth, so, codim W ≥ 2. Consider the composition of maps W ⊆ Y × Z → Y . Let this map be denoted by f . Let V be the non-empty open set of Y such that for all y ∈ Y , f −1 (y) is empty, or has dimension = dim W − dim Y . Suppose all the fibers of points of V is empty. Then, ({y} × Z) ⊆ U . Therefore, F |{y}×Z , G|{y}×Z , (F/G)|{y}×Z are vector bundles and since F |{y}×Z → (F/G)|{y}×Z is surjective, so, kernel of this map is also a vector bundle. But, there is a surjective morphism from G|{y}×Z to the kernel. Since, both are vector bundles of same rank, so, it is an isomorphism. Suppose all fibers of points of V has dimension =dim W −dim Y . Then, dim ({y} × Z)− dim ({y} × Z) ∩ W ≥ 2. Now, the same argument as above shows that G|{y}×Z → F |{y}×Z is injective outside ({y} × Z) ∩ W . Lemma 3.2.3. Let Y be a smooth variety over k. Let F be a coherent sheaf and L a line bundle over C × Y . Suppose there is a non-zero map f : F → L. Then, for a generic choice of y ∈ Y , f |C×{y} is non-zero. Proof. Since, f is non-zero, Image(f ) ⊆ L is non-zero and hence, torsion-free. So, Image(f )|C×{y} = 0 ∀y ∈ Y . Therefore, it is enough to prove that if f is an inclusion, then for a generic choice of y ∈ Y , f |C×{y} is an inclusion. Also, tensoring f by L−1 , we can assume L = OC×Y and F = IZ i.e. the ideal sheaf of a closed subscheme Z ⊆C ×Y. Suppose codim Z = 1. Then, since C × Y is smooth, so IZ is a line bundle. Now, consider the composition of maps g : Z ⊆ C × Y → Y . There exists a non-empty open set V ⊆ Y such that fiber of each point in V is empty or has dim = dim Z− dim Y = 0. If fiber over each point of V is empty, then (C × V ) ∩ Z is empty, so, IZ |C×V = OC×V . Hence, for each point v ∈ V , IZ |C×{v} = OC×{v} . In particular, it is injective. Now, suppose fiber over each point of V has dimension 0. Then for any v ∈ V , Z∩(C×{y}) is a divisor on C. So, ker(OC×{y} → OZ∩(C×{y}) ) is a line bundle on C, and there is a surjection from IZ |C×{y} to this kernel. Hence, it is an isomorphism. If codim Z ≥ 2, then the image of Z in Y is a proper closed subset of Y , so there exists a non-empty open set V in Y such that fiber over each point is empty. Then, following the same argument as above, we can see that for any v ∈ V , IZ |C×{y} = OC×{y} . Theorem 3.2.4. Suppose d ≥ 2. Then if (i) r(r − 1)ab + (d + r − rd)bc + (d − r)ac ≥ 0 (ii) da + (d + r − rd)b ≥ 0 then, F(r, d) is semistable. If each of inequalities are strict, then F(r, d) is stable. if d=1, then F(r, d) is semistable iff rab − bc − ac ≥ 0, and stable iff the inequality is strict. Proof. Let 0 = H ⊆ F (r, d) such that F(r, d)/H is torsion-free. Then, there are three cases that we need to consider: ∗ (a) H πi,P r−1 Ω(1) ∀1 ≤ i ≤ d.
12
CHANDRANANDAN GANGOPADHYAY
∗ (b) H ⊆ πi,P r−1 Ω(1) ∀1 ≤ i ≤ d. ∗ ∗ (c)H ⊆ πi,Pr−1 Ω(1) 1 ≤ i ≤ j and H πi,P r−1 Ω(1) j + 1 ≤ i ≤ d, for some 1 ≤ j ≤ d − 1. ∗ (a) If H πi,P r−1 Ω(1) ∀1 ≤ i ≤ d, then by lemma 2.2, we have
deg H ≤ (−dac − bc)
(3.9)
(−dac − bc) r−1
(3.10)
Hence, μ (H) ≤ Therefore,
−dbc − dac −dac − bc − ) r r−1 (3.11) c = ((r + d − rd)b + da) ≥ 0 r(r − 1) The last inequality is by hypothesis (ii) of the theorem in the case d ≥ 2. If d = 1, then the inequality is automatically satisfied. Hence, μ (F(r, d)) ≥ μ (H). μ (F(r, d)) − μ (H) ≥ (
(b)
d i=1
∗ πi,P r−1 Ω(1) has rank r − d if r ≥ d + 1 and 0 otherwise. So, this case can happen
only when r ≥ d + 1.Therefore, assume r ≥ d + 1. d ∗ πi,P So, if H ⊆ r−1 Ω(1), then rank H ≤ r − d. i=1
Now, using lemma 2.2, we have deg H ≤ −dab(r − 1)
(3.12)
−dab(r − 1) r−d
(3.13)
So, μ (H) ≤ Then,
−dbc − dca −dab(r − 1) − ) r r−d (3.14) d[r(r − 1)ab − (r − d)bc − (r − d)ac] = r(r − d) Now, if d = 1, then r(r − 1)ab − (r − 1)bc − (r − 1)ac ≥ 0 is exactly the hypothesis of the theorem. If d ≥ 2, then the hypothesis (i) of the theorem implies rab(r − 1) − (r − d)ac − (r − d)bc ≥ (dr − d − r)bc − (r − d)bc = (dr − 2r)bc ≥ 0. Hence μ (F(r, d)) ≥ μ (H). r ∗ (c)Just like in case (b), if r ≤ d, then 1 ≤ j ≤ r − 1, since πi,P r−1 Ω(1) = 0. If d ≤ r + 1, μ (F(r, d)) − μ (H) ≥ (
i=1
then 1 ≤ j ≤ d − 1.Also, rank H ≤ r − j. Then, deg H ≤ (−jab(r − 1) − bc − (d − j)ac) μ (H) ≤ Then,
(−jab(r − 1) − bc − (d − j)ac) r−j
(3.15)
STABILITY OF SHEAVES OVER QUOT SCHEMES
−dbc − dac (−jab(r − 1) − bc − (d − j)ac) − ) r r−j (r(r − 1)jab + (−rd + dj + r)bc + (dj − rj)ac) = r(r − j) (j(r(r − 1)ab + dbc + (d − r)ac) + (r − rd)bc) = r(r − j)
13
μ (F(r, d)) − μ (H) ≥ (
(3.16)
If d ≥ 2, by hypothesis (ii), we have r(r − 1)ab + dbc + (d − r)ac + (r − rd)bc ≥ 0. So, r(r − 1)ab + dbc + (d − r)ac ≥ bc(rd − r) ≥ 0. Hence, for any j ≥ 1, j(r(r − 1)ab + dbc + (d − r)ac) + (r − rd)bc ≥ 0. Hence, μ (F(r, d)) ≥ μ (H). (This case cannot happen for d = 1.) Hence, F(r, d) is semistable. Clearly, if each of the inequalities is strict, then F(r, d) is stable. ∗ For d = 1, the inequality rab−bc−ac ≥ 0 is equivalent to the inequality μ (π1,P r−1 Ω(1)) ≤ μ (F(r, 1))). This follows from case (b). So, it is also a necessary condition.
3.3 Semistability and Stability of A(r, d) For a, b > 0, let (πC∗ OC (P ))a C × Q(r, d). Then, we have
∗ πQ (det ((πQ )∗ (B(r, d)
πC∗ OC (lP )))b ) be ample on
Theorem 3.3. (i) Suppose d ≥ 2. Then, if da + (d + r − rd)bl ≥ 0, then with respect to the above ample line bundle, A(r, d) is semistable, and if the inequality is strict, it is stable. (ii) If d = 1, then if ral − bl − a ≥ 0, then A(r, d) is semistable. If the inequality is strict, then it is stable. Proof. (i) Using Corollary 2.3, it is enough to show that F(r, d) is semistable with respect to the polarisation (a, bl, b). By Theorem 3.2.4 , we get that if (a) r(r − 1)abl + (d + r − rd)b2 l + (d − r)ab ≥ 0 (b) da + (d + r − rd)bl ≥ 0 Then, F(r, d) is semistable. Now, it is easy to see that the inequality (b) implies (a). Similarly, all other cases follow from Theorem 3.2.4.
4. Stability of direct image of A(r, d) If det((πQ )∗ (B(r, d)
πC∗ OC (lP ))) is ample on Q(r, d), then
14
CHANDRANANDAN GANGOPADHYAY
Theorem 4.1. (πQ )∗ A(r, d) is stable with respect to the above polarisation on Q(r, d). Lemma 4.2 If r ≥ d + 1, then (πQ )∗ A(r, d) is a torsion-free sheaf of rank r − d. If r ≤ d, then (πQ )∗ A(r, d) = 0.
Proof. The rank of (πQ )∗ A(r, d) is minimum of dimension of H 0 (C × {q}, A(r, d)|C×{q} ) as q varies over Q(r, d). This follows from semi-continuity theorem. Now, for any q ∈ Q(r, d), we have the exact sequence: A(r, d)|C×{q}
0
r OC×{q}
B(r, d)|C×{q}
0
Using the long exact sequence of cohomology, we get: H 0 (C, A(r, d)|C×{q} )
0
kr
kd
Hence, r − d ≤ dim H 0 (C, A(r, d)|C×{q} ) ≤ r. Now, if r ≥ d + 1, then consider any surjection OCr → OCd and consider the compostion of d kc , for some point c ∈ C, where the map is given by restriction this map with OCd → i=1
of each copy of OC to kc . Then, such OCr →
d i=1
kc taking H 0 , we get a surjection
k r → k d . Hence, its kernel has dimension r − d. Therefore, minimum dimension of H 0 (C × {q}, A(r, d)|C×{q} ) as q varies over Q(r, d) is r − d. Hence, rank of A(r, d) is r − d. r d−r kc kcj , where c, cj are all distinct, and If r ≤ d, consider the morphism OCr → the map OCr →
r i=1
i=1
j=1
kc when restricted to c gives an ismorphism of vector spaces, and
1 ≤ j ≤ d − r, let OCr → kcj is given by any surjection. Then, taking global sections of r d−r OCr → kc kcj , we get an inclusion k r → k d . Hence, minimum rank of A(r, d)|C×{q} i=1
j=1
r is 0. Since, (πQ )∗ A(r, d) ⊆ OQ , so, this implies (πQ )∗ A(r, d) = 0.
Remark : From now on, we will assume r ≥ d + 2, since, if r ≤ d then (πQ )∗ A(r, d) = 0, and if r = d + 1, then it is a rank 1 sheaf.
Recall that we have the following diagram:
STABILITY OF SHEAVES OVER QUOT SCHEMES
C ×U
idC ×Φ
C × Q(r, d)
π2
U
15
πQ Φ
Q(r, d)
Lemma 4.3. The natural map Φ∗ (πQ )∗ A(r, d) → (π2 )∗ (idC × Φ)∗ A(r, d) is an isomorphism outside a closed subset of codimension ≥ 2.
Proof. For any permutation (i1 , i2 , .., id ) of (1, 2, .., d), define Zi1 ,i2 ,..,id = {u ∈ U|pid ,Pr−1 (u) ∈ subspace generated bypi1 ,Pr−1 (u), pi2 ,Pr−1 (u), .., pid−1 ,Pr−1 (u) in Pr−1 }. Since r ≥ d + 2, Zi1 ,i2 ,..,id is a closed subset of codim ≥ 2. Let U = U \ Zi1 ,i2 ,..,id .
i1 ,i2 ,..,id (r−1) : ... : vi ] be a choice of Let u = ((c1 , v1 ), (c2 , v2 ), .., (cd , vd )) ∈ U . Let vi = d r ∗ → πi,P homogeneous coordinate for vi . Then, the quotient OC×U r−1 O(1)|0,i,C on i=1 d kci , where OCr → kci is given C × U , when restricted to C × {u} becomes OCr → i=1 (0) (1) (r−1) . Hence, the map by the element vi ∈ Pr−1 ,i.e. e(0) → vi , e(1) → vi , .., e(r−1) → vi ⎡ (0) (1) (r−1) ⎤ v1 v1 ... v1 (0) (1) (r−1) ⎥ ⎢ v2 v2 ... v2 d ⎢ ⎥ ⎥. H 0 (C, OCr ) → H 0 (C, kci ) is given by the matrix ⎢ ⎢ . ⎥ i=1 ⎣ ⎦ (0) [vi
.
(0) vd
(1)
(r−1)
vd ... vd Since, by definition of U , the set {v1 , v2 , ..., vd } is linearly independent, i.e. the rows of the above matrix are linearly independent. Therefore, rank of the above matrix is d and d hence, the morphism H 0 (C, OCr ) → H 0 (C, kci ) is surjective. i=1
Now, by Grauert’s theorem [3, Chapter III,Corollary 12.9 ], we have the following commutative diagram:
r )|u (π2 )∗ (OC×U
(π2 )∗ (
d i=1
∗ πi,P r−1 O(1)|0,i,C )|u ∼ =
∼ =
H 0 (C, OCr )
H 0 (C,
d i=1
r Therefore, the map (π2 )∗ (OC×U )|u → (π2 )∗ (
we have the following commutative diagram:
k ci )
d i=1
∗ πi,P r−1 O(1)|0,i,C )|u is surjective. Now,
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CHANDRANANDAN GANGOPADHYAY
(π2 )∗ F(r, d)|u
r (π2 )∗ (OC×U )|u
(π2 )∗ (
d i=1
∗ πi,P r−1 O(1)|0,i,C )|u ∼ =
∼ =
0
H 0 (C, F(r, d)|C×{u} )
0
r H 0 (C, OC×{u} )
H 0 (C,
d i=1
k ci )
0
Here, the rows are exact and hence, the map (π2 )∗ F(r, d)|u → H 0 (C, F(r, d)|C×{u} ) is surjective. Then, using [3, Chapter III, Theorem 12.11] , it is an isomorphism. d r r r → B(r, d)|C×{Φ(u)} = OC×{u} → kci . So,H 0 (C, OC×{Φ(u)} )→ Now, for Φ(u), the map OC×{Φ(u)} i=1
H 0 (C, B(r, d)|C×{Φ(u)} )) is again surjective. Therefore, applying the same argument as above, we get that (πQ )∗ A(r, d)|Φ(u) ∼ = H 0 (C, A(r, d)|C×{Φ(u)} ). Therefore, we have: Φ∗ (πQ )∗ A(r, d)|u = (πQ )∗ A(r, d)|Φ(u)
(π2 )∗ F(r, d)|u ∼ =
∼ =
H 0 (C, F(r, d)|C×{u} ) Hence, ∀ u ∈ U , Φ∗ (πQ )∗ A(r, d)|u → (π2 )∗ (idC ×Φ)∗ A(r, d)|u is an isomorphism. There fore, Φ∗ (πQ )∗ A(r, d)|U → (π2 )∗ (idC × Φ)∗ A(r, d)|U is an isomorphism.
Corollary 4.4. If (π2 )∗ F(r, d) is stable with respect to the polarisation (p∗i,C OC (l) p∗i,Pr−1 OPr−1 (1)), then (πQ )∗ A(r, d) is stable with respect to i=1 det(π∗ (B(r, d) πC∗ OC (l))). d
Proof. This follows from Lemma 1.1, Theorem 2.1(iii) and Lemma 4.3.
Proof of theorem 4.1. By the above corollary, it is enough to show that (π2 )∗ F(r, d) d over X = (C × Pr−1 )d is stable with respect to (p∗i,C OC (b) p∗i,Pr−1 OPr−1 (c)), for any b, c > 0. Now, we have the following left exact sequence:
0
(π2 )∗ F(r, d)
OXr
i=1
(π2 )∗ (
d i=1
∗ πi,P r−1 O(1)|0,i,C )
STABILITY OF SHEAVES OVER QUOT SCHEMES
Claim: (π2 )∗ ( OXr → (π2 )∗ (
d i=1
d i=1
17
∗ ∼ p∗ r−1 O(1) and the map πi,P r−1 O(1)|0,i,C ) = i,P d
i=1
d ∗ ∼ p∗ r−1 O(1) is such that the projection to i-th facπi,P r−1 O(1)|0,i,C ) = i,P i=1
tor, i.e. the map OXr → p∗i,Pr−1 O(1) is given by p∗i,Pr−1 (OPr r−1 → O(1)). proof of claim. Now, the map d d ∗ r ∗ (π2 )∗ (πi,P OXr → (π2 )∗ ( πi,P r−1 O(1)|0,i,C ) = OX → r−1 O(1)|0,i,C ). i=1
i=1
∗ ∼ ∗ So, it is enough to show (π2 )∗ π1,P r−1 )O(1)|0,i,C = p1,Pr−1 O(1) and r ∗ ∗ r (π2 )∗ (OC×X → π1,P r−1 O(1)|0,1,C ) = p1,Pr−1 (OPr−1 → O(1)).
Now, 0,1,C = X and if, j :0,1,C → C ×X is the inclusion, then, π2 ◦j is the isomorphism and π1,Pr−1 ◦ j = p1,Pr−1 . ∗ ∗ ∗ ∗ So, (π2 )∗ (π1,P r−1 O(1)|1,i,C ) = (π2 )∗ j∗ (j πi,Pr−1 O(1)) = pi,Pr−1 O(1). r ∗ Also, the map OC×X → π1,P r−1 O(1)|0,1,C factors as : ∗ π1,P r−1 O(1)|0,1,C
Or 0,1,C
r OC×X
where the second map is just j∗ (OXr → p∗1,Pr−1 O(1)), so, (π2 )∗ (j∗ )(OXr → p∗1,Pr−1 O(1)) gives r the required morphism. The only thing we need to show is that (π2 )∗ (OC×X → Or 0,1,C ) is r the identity map from OXr → OXr . Now, (π2 )∗ (Or 0,1,C ) = OXr , and H 0 (C × X , OC×X )→ 0 r H (C × X , O0,1,C ) is identity. Hence, the result follows. Proof of theorem 4.1(contd.). Using the claim, we get that d p∗i,Pr−1 Ω(1) ⊆ OXr . (π2 )∗ F(r, d) = i=1
Now, by the proof of the Lemma 4.2, we get that the following sequence is exact outside a codimension ≥ 2 subset:
(π2 )∗ F(r, d)
0
d
OXr
i=1
p∗i,Pr−1 O(1)
Therefore, using Corollary 3.1.2, we get: d (rd − 1)!(bd )(cr−1 )d−1 cr−2 (−d). deg (π2 )∗ F(r, d) = − deg p∗i,Pr−1 O(1) = ((r − 1)!)d−1 (r − 2)! i=1 d p∗i,Pr−1 Ω(1) such that the quotient is torsion free. Then, using Lemma Now, let H ⊆ i=1
3.2.2, we have H|S1,Pr−1 → ( of Pr−1 . Also, note that
d i=1
d i=1
p∗i,Pr−1 Ω(1))|Si,Pr−1 outside a codimension ≥ 2 closed subset
p∗i,Pr−1 Ω(1) ⊆
d−1 i=1
p∗i,Pr−1 Ω(1) is such that the quotient is torsion
18
CHANDRANANDAN GANGOPADHYAY
free. This follows from the following exact sequence: d
0
i=1
Similarly, we have H ⊆
d−1
p∗i,Pr−1 Ω(1)
d i=1
i=2
p∗i,Pr−1 Ω(1) ⊆
d−1 i=1
p∗i,Pr−1 Ω(1)
p∗i,Pr−1 Ω(1) ⊆
p∗d,Pr−1 O(1)
d−2 i=1
(4.1)
p∗i,Pr−1 Ω(1) ⊆ ... ⊆ p∗1,Pr−1 Ω(1)
such that each successive quotient is torsion free. Hence, H|S1,Pr−1 → p∗1,Pr−1 Ω(1) is injective outside a codimension ≥ 2 closed subset of Pr−1 |S1,Pr−1 . So, deg H|S1,Pr−1 ≤ −1. Same is true for any i. Therefore, using Corollary 3.1.2, we get that deg H ≤ d d (rd − 1)!bd (cr−1 )d−1 cr−2 ∗ (−d) = deg p Ω(1).Since, rank H < rank p∗i,Pr−1 Ω(1), r−1 i,P ((r − 1)!)d−1 i=1 i=1 d d p∗i,Pr−1 Ω(1) is stable. hence, μ(H) < μ( p∗i,Pr−1 Ω(1)). Hence, i=1
i=1
5. Stability of sheaves over flag schemes Fix d = (d1 , d2 , .., dk ) ∈ Nk with d1 d2 .. dk 0 and let r ≥ 1. Let Q(r, d) be scheme whose closed points correspond to filtered quotients OCr → B1 → B2 → .. → Bk such that each OCr → Bj is surjective and Bj is a torsion sheaf of length dj for 1 ≤ j ≤ k. Then, over Q(r, d) we have chain of quotients: r OC×Q(r,d)
B1 (r, d)
B2 (r, d)
...
Bk (r, d)
r Let us denote the kernel of OC×Q(r,d) → Bj (r, d) by Aj (r, d). Let πC and πQ be the projections from C × Q(r, d) to C and Q(r, d) respectively. k Then, det (πQ )∗ (Bj (r, d) π ∗ OC (lj )) is ample for lj >> 0 ∀j. Let us fix such an j=1
ample line bundle and let it be denoted by L. Then, we have the following theorem: Theorem 5.1. (πQ )∗ Ai (r, d) is stable with respect to L. Proof. Let d1 = d. Following the notation of the previous sections, we have X = (C × d r ∗ Pr−1 ). Recall that over C × X we have the morphism q : OC×X → πi,P r−1 O(1)|0,i,C . i=1
Now, consider the chain of morphisms over C × U , r OC×X
q
→ −
d i=1
∗ πi,P r−1 O(1)|0,i,C
→
d2 i=1
∗ πi,P r−1 O(1)|0,i,C
→ ... →
dk i=1
∗ πi,P r−1 O(1)|0,i,C
(5.1)
STABILITY OF SHEAVES OVER QUOT SCHEMES
where
dj i=1
∗ πi,P r−1 O(1)|0,i,C →
d j+1 i=1
19
∗ πi,P r−1 O(1)|0,i,C is just the projection to the first dj+1
factors. Now, by theorem 2.1, q|C×U is a surjection, hence, each of the morphisms dj r ∗ OC×X → πi,P r−1 O(1)|0,i,C in the above chain, when restricted to C × U is surjeci=1
tive. Therefore, by universal property of Q(r, d) we have a map Φd : U → Q(r, d) such that dj r r ∗ → Bj (r, d)) = (OC×X → πi,P (i) Φ∗d (OC×Q(r,d) r−1 O(1)|0,i,C )|C×U i=1
r (ii)(idC × Φd )∗ A(r, d) = ker (OC×X →
dj
i=1
∗ πi,P r−1 O(1)|0,i,C )|C×U
d (iii)(idC × Φd )∗ L ∼ = ( (p∗i,C O(ai ) p∗i,Pr−1 OPr−1 (bi )))|U for some ai , bi > 0. i=1
(iv)(a) If dj = r + 1, then rank of (πQ )∗ Aj (r, d) is 1. If dj ≤ r, then (πQ )∗ Aj (r, d) = 0. (b)If dj ≤ r + 2, then (Φd )∗ (πQ )∗ Aj (r, d) ∼ = (π2 )∗ (idC × Φd )∗ Aj (r, d) outside a closed subset of X of codimension ≥ 2. The proofs of (i), (ii) and (iii) are same as in Theorem 2.1. The proof of (iv)(a) is same as in Lemma 4.2 and the proof of (iv)(b) is same as in Lemma 4.3. By (iv)(a), if r ≤ d + 1,then we are done. So, let r ≥ dj + 2. dj r ∗ πi,P Let us denote ker (OC×X → r−1 O(1)|0,i,C ) by Fj (r, d). i=1
Then, using Lemma 1.1 , (iii) and (iv)(b) it is enough to show that (π2 )∗ Fj (r, d) is stable d with respect to the polarisation (p∗i,C O(ai ) p∗i,Pr−1 OPr−1 (bi )) for any ai , bi > 0. i=1
Now, by the claim in the proof of Theorem 4.1. , (π2 )∗ Fj (r, d) = rest of the proof is same as in Theorem 4.1.
dj i=1
p∗i,Pr−1 Ω(1). Now, the
20
CHANDRANANDAN GANGOPADHYAY
References [1] [2] [3]
D. Huybrechts and M. Lehn, The Geometry of Moduli Spaces of Sheaves, Cambridge University Press, Second Edition(2010). V. Balaji, L. Brambila-Paz and P. E. Newstead, Stability of the Poincare bundle, Math. Nachr. 188 (1997), 5–15. R. Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, (1977).
School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India E-mail address: [email protected]
Chandranandan Gangopadhyay
PII: DOI: Reference:
S0007-4497(18)30075-7 https://doi.org/10.1016/j.bulsci.2018.08.001 BULSCI 2761
To appear in:
Bulletin des Sciences Mathématiques
Received date:
1 June 2018
Please cite this article in press as: C. Gangopadhyay, Stability of sheaves over Quot schemes, Bull. Sci. math. (2018), https://doi.org/10.1016/j.bulsci.2018.08.001
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STABILITY OF SHEAVES OVER QUOT SCHEMES CHANDRANANDAN GANGOPADHYAY
Abstract. Let C be a smooth projective curve over an algebraically closed field k. Let r Q(r, d) be the Quot scheme of degree d zero dimensional quotients of OC . Let A(r, d) be the kernel of the universal quotient map over C × Q(r, d) and let πQ be the second projection from C × Q(r, d) to Q(r, d). Then, we prove that with respect to certain natural polarisations on C × Q(r, d) and Q(r, d), A(r, d) and (πQ )∗ A(r, d) are stable. A r is also proved. similar statement over Flag scheme of torsion quotients of OC
1. INTRODUCTION Fix r, d ≥ 1. Let C be a smooth, projective curve over an algebraically closed field k. Let Q(r, d) be the Quot scheme of degree d zero dimensional quotients of OCr . Over C × Q(r, d), we have the universal exact sequence: 0
A(r, d)
B(r, d)
r OC×Q(r,d)
0
Let πC and πQ are the projections from C × Q to C and Q respectively. Then, if P ∈ C, then OC (P ) is ample, and det((πQ )∗ (B(r, d) πC∗ OC (lP ))) is ample on Q(r, d) for l >> 0. This follows from [1, Chapter 2, Proposition 2.2.5]. Then, we prove that: Theorem 4.1. With respect to the above polarisation, the coherent sheaf (πQ )∗ A(r, d) is stable. Also, for a, b > 0, the line bundle (πC∗ OC (P ))a is ample on C × Q(r, d). Then, we prove that
∗ (det ((πQ )∗ (B(r, d) πQ
πC∗ OC (lP )))b )
Theorem 3.3. (i) Suppose d ≥ 2. Then, if da + (d + r − rd)bl ≥ 0, then with respect to the above ample line bundle A(r, d) is semistable, and if the inequality is strict, it is stable. (ii) If d = 1, then if ral − bl − a ≥ 0, then A(r, d) is semistable. If the inequality is strict, then it is stable. Infact, the analogous statement to Theorem 4.1. in the case of Flag schemes of filtrations of torsions quotients of OCr is also true [Theorem 5.1]. The proof is essentially the same as in the case of Quot schemes. We give an outline of the proof in the fifth section. 1
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CHANDRANANDAN GANGOPADHYAY
Note that if r = 1, then, A(r, d) ⊆ OC×Q(r,d) and (πQ )∗ A(r, d) ⊆ (πQ )∗ OC×Q(r,d) = OQ(r,d) , and hence, they have rank 1 or 0. Therefore, they are stable. So, from here onwards, we assume that r ≥ 2. We will be using the following lemma [2, Lemma 1.1] to reduce the problem to checking semistability(or stability) of certain sheaves over C × (C × Pr−1 )d and (C × Pr−1 )d : Lemma 1.1. [2, Lemma 1.1] Let X and Y are two smooth projective varieties of same dimension. Let f : X Y be a dominant rational map defined outside a subset Z ∈ X of codim ≥ 2. Let DX and DY be ample divisors on X and Y respectively such that f ∗ DY |X\Z = DX |X\Z . Let E be a vector bundle on Y such that f ∗ E extends to a vector bundle F on X. Then, if F is DX -semi-stable(resp. stable), then E is DY -semi-stable (resp. stable). To use the above lemma, we first show that there exists a rational map Φ : (C × Pr−1 )d Q(r, d), which is defined outside the closed subset of codimension 2.
2. Construction of the map Φ Notation: Let X = (C × Pr−1 )d , let the i-th projection to Pr−1 be denoted by pi,Pr−1 and the i-th projection to C be denoted by pi,C . For {i1 , i2 , ..., ik } ⊆ {1, 2, ..., d}, k ≥ 2, let i1 ,i2 ,...,ik = {x ∈ X |pi1 ,C (x) = pi2 ,C (x) = ... = pik ,C (x), pi1 ,Pr−1 (x), pi2 ,Pr−1 (x).., pik ,Pr−1 (x) are linearly dependant}. Let Z = i1 ,i2 ,...,ik and let U = X \Z. Note that each of these i1 ,i2 ,..,ik has codimension ≥ 2 and hence, Z has codimension 2 in X . Let π1 and π2 be the first and second projection from C × X to C and X respectively, and let pi,C ◦ π2 be denoted by πi,C and pi,Pr−1 ◦ π2 be denoted by πi,Pr−1 . Now, over C × X , let q be the following map:
q:
r OC×X
→
d
∗ πi,P r−1 O(1)|0,i,C
(2.1)
i=1 r ∗ where 0,i,C = {(c, x) ∈ C × X |c = πi,C (x)} and the quotient OC×X → πi,P r−1 O(1)|0,i,C is r−1 r , O → O(1) by πi,Pr−1 followed obtained by pulling back the natural quotient over P Pr−1 by the restriction to 0,i,C .
Theorem 2.1. There exists a dominant map Φ : U → Q(r, d) such that: r (i) (idC × Φ)∗ (OC×Q(r,d) → B(r, d)) = q|C×U ∗ ∼ (ii) (idC × Φ) A(r, d) = ker q|C×U d (iii) Φ∗ det ((πQ )∗ (B(r, d) πC∗ OC (lP )))= (p∗i,C OC (lP ) p∗i,Pr−1 OPr−1 (1))|U . i=1
STABILITY OF SHEAVES OVER QUOT SCHEMES
3
Proof. (i) By the universal property of Quot schemes, it is enough to show that q|C×U is ∗ surjective, and di=1 πi,P r−1 O(1)|0,i,C , when restricted to C × {x}, for x ∈ X , is a torsion sheaf of degree d. d ∗ If x = ((c1 , v1 ), (c2 , v2 ), .., cd , vd ) ∈ X , then πi,P r−1 O(1)|1,i,C when restricted to C × {x} is nothing but
d i=1
i=1
kci , hence it is of degree d.
Claim: q|C×U is a surjection.
r Proof. Now let x = ((ci , vi )) ∈ U . Then, over C×{x}, q|C×{x} is given by OC×{x} →
d i=1
k ci ,
where, each OCr → kci is given by vi , i.e. if we choose homogeneous co-ordinate of vi = (0) (1) (r−1) (0) (1) [vi , vi , .., vi ], then, the map OCr → kci is given by: e(0) → vi , e(1) → vi , .., e(r−1) → (r−1) . vi We only need to check surjectivity at the points c1 , c2 , .., cd . Without loss of generality, assume c1 = c2 = .. = cj = c and c = ci ∀i ≥ j + 1. Then, q|{c}×{x} is given by the map: k r → k j , with (0) (0) (0) (1) (1) (1) (r−1) (r−1) (r−1) , v2 , .., vj ), i.e. e0 → (v1 , v2 , .., vj ), e1 → ((v1 , v2 , .., vj )),..., er−1 → (v1 ⎡ ⎤ (0) (1) (r−1) v v .. v1 ⎢ 1(0) 1(1) (r−1) ⎥ ⎢v ⎥ v2 .. v2 the matrix of this map is given by: ⎢ 2 ⎥ Now, since x ∈ U , hence, the . .. .. .. ⎣ ⎦ (0) (1) (r−1) vj vj .. vj rows are linearly independant. Therefore, rank of the above matrix is j. Hence, the above map is surjective at c. To see that Φ is dominant, first note that if x ∈ Q(r, d) corresponds to a quotient d kci with all ci ’s distinct, then, Φ−1 (x) is a non-empty finite set. Therefore, OCr → i=1
dim Image Φ = dim X = rd. Now, since, Q(r, d) is a smooth projective variety of dimension rd, so, Image Φ = Q(r, d). (ii) Note that there is a surjection from (idC × Φ)∗ A(r, d) → (ker q)|C×U . So, the claim follows from the fact that both of these sheaves are vector bundles of same rank. This fact is proved in Lemma 1.1. (iii) This is proved in two steps: ∗ Step 1: Φ (πQ )∗ (B(r, d) πC∗ O(lP )) ∼ = (π2 )∗ (idC × Φ)∗ (B(r, d) πC∗ O(lP )). d Step 2: (π2 )∗ (idC × Φ)∗ (B(r, d) πC∗ O(lP )) ∼ = (p∗i,C OC (lP ) p∗i,Pr−1 OPr−1 (1))|U . i=1
Then, (iii) will follow by taking the top exterior powers of the above bundles. Proof of Step 1: Let G = B(r, d) πC∗ O(lP ). We have the natural map Φ∗ (πQ )∗ (G) →
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CHANDRANANDAN GANGOPADHYAY
(π2 )∗ (idC × Φ)∗ (G) such that if u ∈ U , then we have the following commutative diagram: (Φ∗ (πQ )∗ G)|u = ((πQ )∗ G)|Φ(u)
((π2 )∗ (idC × Φ)∗ G)|u H 0 (C, G|C×{u} )
Since, for all x ∈ Q(r, d), G|C×{x} is a torsion sheaf of degree d, hence, by [3, Chapter III, Corollary 12.9], the map ((πQ )∗ G)|x → H 0 (C, G|C×{u} ) is an isomorphism. Similarly,the vertical arrow in the above diagram is also an isomorphism. Hence (Φ∗ (πQ )∗ G)|u → ((π2 )∗ (idC × Φ)∗ G)|u is an isomorphism. Therefore (Φ∗ (πQ )∗ G) → ((π2 )∗ (idC × Φ)∗ G) is an isomorphism. Proof of Step 2: d d ∗ ∗ ∗ ∗ π1 OC (lP )) = (π2 )∗ (πi,P π1 OC (lP )). (π2 )∗ ( πi,P r−1 O(1)|0,i,C r−1 O(1)|0,i,C =
d i=1
i=1
∗ (π2 )∗ (πi,P r−1 O(1)|0,i,C
i=1
π1∗ OC (lP )|0,i,C ).
Now, 0,i,C = X and if j :0,i,C → C × X is the inclusion, then π2 ◦ j is identity, π1 ◦ j = pi,C . Hence, πi,Pr−1 ◦ j = pi,Pr−1 and ∗ ∗ ∗ ∗ O(1)| j π1 OC (lP )) π1∗ OC (lP )|0,i,C ) = (π2 )∗ (j∗ )(j ∗ πi,P (π2 )∗ (πi,P r−1 r−1 O(1) 0,i,C ∗ ∗ = pi,Pr−1 O(1) pi,C OC (lP ).
Lemma 2.2. Let Y be a variety over k. Over C ×Y , suppose we are given a surjection; r → F such that for any y ∈ Y , F|C×{y} is a torsion sheaf over C of degree d. f : OC×Y r is an injection. Then, ker f is a vector bundle of rank r and (ker f )|C×{y} → OC×{y} Proof. Since, for all y ∈ Y , the Hilbert polynomial of F|C×{y} with respect to any polarir is also sation is the constant polynomial d, it follows that F is flat over Y . Since, OC×Y flat over Y so, ker f is also flat over Y . Now, flatness of ker f implies for all y ∈ Y , (ker f )|C×{y} ∼ = ker (f |C×{y} ). This alr ready implies the injectivity of ker f )|C×{y} → OC×{y} . Also this sheaf, being a subsheaf of OCr , is locally free, and since its cokernel is torsion, it has rank r. Hence, for all (c, y) ∈ C × Y , (ker f )|{c}×{y} is a vector space of rank r. Since, the rank remains constant for all (c, y) ∈ C × Y , this implies that ker f is a vector bundle.
Notation: Let us denote ker q by F(r, d). Corollary 2.3. If F(r, d) is semistable with respect to the polarisation d ∗ ∗ π1∗ OC (a) (πi,C OC (lb) πi,P r−1 OPr−1 (b)), then A(r, d) is semistable with respect to i=1 ∗ (det ((πQ )∗ (B(r, d) πC∗ OC (lP )))b ). (πC∗ OC (P ))a πQ
STABILITY OF SHEAVES OVER QUOT SCHEMES
5
[For notational convenience, we denote OC (lP ) by OC (l).]
Proof. Since, Q(r, d) is a smooth projective variety of dimension rd = dim X , so this corollary follows from Lemma 1.1 and Theorem 2.1. 3. Semistability and Stability of A(r, d) From now on, we will denote by (a, b, c) the polarisation on C × X corresponding to the d ∗ ∗ πi,C O(b) πi,P line bundle π1∗ O(a) r−1 O(c). First, we will determine a condition on i=1
(a, b, c) such that F(r, d) is semistable(or stable ) with respect to (a, b, c)[Theorem 3.2.4] and then use Corollary 2.3 to find polarisations on C × Q(r, d) with respect to which A(r, d) is semistable(or stable). 3.1 Degree of Sheaves with respect to the polarisation (a,b,c)
Theorem 3.1.1. Let X1 , X2 , ..., Xn are smooth projective varieties of dimension d1 , d2 , ..., dn respectively. Let DXi be an ample divisor on Xi . If pi : X1 ×X2 ×...×Xn → Xi n is the i-th projection to Xi , then p∗i DXi is an ample divisor of X1 × X2 × .. × Xn . Let i=1
L be a line bundle on X1 × X2 × .. × Xn . Then,with respect to deg L =
n
i=1
n
(d1 + d2 + .. + dn − 1)! deg d1 !d2 !..(di − 1)!dn! j=1,j=i
n i=1
DXj (DXj )
p∗i DXi , deg
DXi L|Si ,
(3.1)
where, if for each i, if we have any choice of xi,j ∈ Xj , j = i, then Si = {x ∈ X1 × X2 × ... × Xn |pj (x) = xi,j , ∀j = i}. Proof. Let Y is a smooth projective variety of dimension m, We denote the chow group of codimension i-cycles on Y by CH i (Y ). Since, Y is smooth, we have the intersection product : CH i (Y ) × CH j (Y ) → CH i+j (Y ). Also, since Y is projective, we have the degree map deg : CH n (Y ) → Z, given by k k [ ni Pi ] → ni , where Pi ∈ Y . i=1
i=1
Let L be a line bundle on Y and let D be an ample divisor on Y . Let us denote the class of L and D in CH 1 (Y ) by [L] and D respectively. Then, degree of L with respect to the ample divisor D is given by : deg([L].[D]m−1 ). Hence, in the context of our theorem, deg L with respect to degree of the following cycle :
n i=1
p∗i DXi , is given by the
6
[L].[
CHANDRANANDAN GANGOPADHYAY n i=1
[
p∗i DXi ](d1 +d2 +..+dn −1) ∈ CH d1 +d2 +..+dn (X1 × X2 × ... × Xn ). Now,
n
p∗i DXi ](d1 +d2 +..+dn −1)
=[
n
(i1 +i2 +..in ) =d1 +d2 +..+dn−1
i=1
=[
(d1 + d2 + ..dn − 1)! ∗ ij pj (DXj )] i1 !i2 !..in ! j=1 n
(i1 +i2 +..in ) =d1 +d2 +..+dn−1
=[
(d1 + d2 + ..dn − 1)! ∗ (pj DXj )ij ] i1 !i2 !..in ! j=1
d n
(d1 + d2 + ..dn − 1)! i=1
d1 !d2 !..(di − 1)!..dn !
(3.2)
di −1 p∗j (DXjj )p∗i (DX )] i d
j=1 j=i
The third equality follows from the fact that, if for some j, ij ≥ dj + 1, then (DXj )ij ∈ CH ij (Xj ) = 0. Hence, it is enough to show that the n n d di −1 degree of [L].[ p∗j (DXjj )p∗i (DX )] = deg i j=1,j=i
j=1 j=i
DXi (DXi )
deg
DXi L|Si .
d
Without loss of generality, assume i = 1. Then, for each 2 ≤ j ≤ n, let [DXjj ] = where xj,l ∈ Xj . Note that, by definition, deg Then, [
n
d1 −1 p∗j (DXjj )p∗1 (DX )] = [ 1 d
j=2
n
n j=2
p∗j (
kj l=1
DXj (DXj )
= kj .
d1 −1 p∗j (xj,l ))p∗1 (DX )] = 1
kj
[xj,l ],
l=1
n d1 −1 [ p∗j (xj,lj )p∗1 (DX )]. 1 lj j=2
d1 −1 p∗j (xj,lj )p∗1 (DX )] = [[(DX1 )d1 −1 ] × {x2,l2 } × {x3,l3 } × .. × {xn,ln }]. This follows 1 j=2 from the fact that if Y and Z are two smooth projective varieties, then CH ∗ (Y ) CH ∗ (Z) → CH ∗ (Y × Z) is a ring homomorphism. Now, by projection formula, [L][[(DX1 )d1 −1 ]×{x2,l2 }×{x3,l3 }×..×{xn,ln }] = [L|X1 ×{x2,l2 }×{x3,l3 }×..×{xn,ln } ].[(DX1 )d1 −1 ] = deg DX1 L|S1 . n d d1 −1 )] = deg DX1 L|S1 = k2 k3 ..kn (deg DX1 L|S1 ). Since, kj = Hence, [L][ p∗j (DXjj )p∗1 (DX 1
Now, [
j=2
deg
DXj (DXj ),
lj
we are done.
Corollary 3.1.2. Under the hypothesis of the above theorem, if F is a coherent sheaf on X1 × X2 × .. × Xn , then: deg F = deg F =
n n
(d1 + d2 + .. + dn − 1)! i=1
d1 !d2 !..(di − 1)!dn!
j=1,j=i
deg
DXi (DXi )
deg
DXi F |Si ,
(3.3)
STABILITY OF SHEAVES OVER QUOT SCHEMES
7
where, for each i, we have a generic choice of xi,j ∈ Xj , j = i, and Si = {x ∈ X1 × X2 × ... × Xn |pj (x) = xi,j , ∀j = i}.
Proof. Since X1 × X2 × .. × Xn is projective, F has a locally free resolution: 0 Fk Fk−1 ... F0 F 0 where, Fi are locally free. k l Then, det F = (det Fl )(−1) and deg F = deg (det F ). l=1
ˆi × .. × Xn such that Now, for each i, there exists a non-empty open set Ui ⊆ X1 × X2 × ..X n F |Ui ×Xi is flat over Ui . Hence, for any point (xi,j )j=1 ∈ Ui , the above resolution restricted j=i
to Si , remains exact, and gives a resolution of F |Si . In other words, det (F |Si )= (det F )|Si . Now, applying the above theorem for the line bundle det F , we get the required result. Corollary 3.1.3. Let G is a coherent sheaf over C × X . Then,
(rd)!bd−1 (cr−1 )d−1 cr−2 [bc deg G| + ac deg G|Si,C deg G = S 0,C ((r − 1)!)d i=1 d
+
d
(3.4)
(ab)(r − 1) deg G|Si,Pr−1 ]
i=1
Here, by deg G|Si,Pr−1 , we mean degree with respect to OPr−1 (1).
Proof. Using the above corollary, the degree of G with respect to the polarisation (a,b,c) is given by :
(rd)! (rd)! d r−1 d b (c ) deg G| + abd−1 (cr−1 )d deg G|Si,C deg G = S 0,C d ((r − 1)!)d ((r − 1)!) i=1 d
d
(rd)! abd (cr−1 )d−1 degc G|Si,Pr−1 d−1 (r − 2)! ((r − 1)!) i=1 (3.5) By degc G|Si,Pr−1 we mean degree of G|Si,Pr−1 with respect to O(c). Let us denote deg1 G|Si,Pr−1 by deg G|Si,Pr−1 . Then, degc G|Si,Pr−1 = cr−2 deg G|Si,Pr−1 . Then, we have +
(rd)!bd−1 (cr−1 )d−1 cr−2 [bc deg G| + ac deg G|Si,C S0,C ((r − 1)!)d i=1 d
deg G =
+
d
i=1
(ab)(r − 1) deg G|Si,Pr−1 ]
(3.6)
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CHANDRANANDAN GANGOPADHYAY
Notation: Let us denote the constant denote the
(rd)!bd−1 (cr−1 )d−1 cr−2 by K. Then, we will ((r − 1)!)d
μ(G) deg G by deg G and by μ (G). K K
Corollary 3.1.4. deg F(r, d) = −d(ac + bc)
Proof. Since the following sequence is exact outside a codimension 2 subset, we can compute deg F(r, d):
F(r, d)
0
i.e. deg F(r, d) = −
d i=1
d
q
r OC×X
i=1
∗ πi,P r−1 O(1)|0,i,C
∗ deg πi,P r−1 O(1)|0,i,C = −d(ac + bc).
(3.7)
3.2 Semistability and Stability of F(r, d)
d
r ∗ → πi,P Over C × X , let us denote ker (OC×X r−1 O(1)|0,i,C ) by Gi . Clearly, F(r, d) =
i=1
Gi .
Now, we have the following commutative diagram:
0 I0,i,C (1) 0
∗ πi,P r−1 Ω(1)
r OC×X
∗ πi,P r−1 O(1)
0
0
Gi
r OC×X
∗ πi,P r−1 O(1)|0,i,C
0
0
STABILITY OF SHEAVES OVER QUOT SCHEMES
9
Using snake lemma we have the following exact sequence: 0
∗ πi,P r−1 Ω(1)
Gi
I0,i,C (1)
0
(3.8)
Now, let H ⊆ F(r, d) be a subsheaf such that F(r, d)/H is torsion free. Then we have the following lemma: Lemma 3.2.1. For generic choices of c, ci ∈ C , vi ∈ Pr−1 , we have (i) deg H|Si,C ≤ 0, deg H|Si,Pr−1 ≤ 0 ∀i ∈ {1, 2, .., d} ∗ If H ⊆ πj,P r−1 Ω(1), then (ii)deg H|Sj,Pr−1 ≤ −1 ∗ If H πj,P r−1 Ω(1), then (iii)deg H|Sj,C ≤ −1, deg H|S0,C ≤ −1. Proof. First, we claim that for generic choices of c, ci ∈ C, vi ∈ Pr−1 , (a)the maps H|Si,C → F(r, d)|Si,C , H|Si,Pr−1 → F(r, d)|Si,Pr−1 , H|S0,C → F(r, d)|S0,C are all injective outside a closed subset of codimension ≥ 2. (b)the maps F(r, d)|Si,C → OCr , F(r, d)|Si,Pr−1 → OPr r−1 , F(r, d)|S0,C → OCr are all injective. Claim(a) will follow from Lemma 3.2.2. Proof of Claim (b). First we prove that for a generic choice of c, cj ∈ C and vi ∈ Pr−1 , F(r, d)|Si,C → OCr is injective. Without loss of generality, assume i = 1. Consider the map C × (C × Pr−1 )d
given by π1 × π1,Pr−1
C × Pr−1 × (C × Pr−1 )d−1 × (π2,C × π2,Pr−1 ) × ... × (πd,C × πd,Pr−1 ).
Consider the open set V ⊆ C × Pr−1 × (C × Pr−1 )d−1 of points of the form (c, v1 , (c2 , v2 ), .., (cd , vd )) with no two of c, ci equal. d r ∗ Then, when we restrict q : OC×X → i=1 πi,Pr−1 O(1)|0,i,C to the fibre of a point (c, v1 , (c2 , v2 ), .., (cd , vd )) ∈ V , it becomes OCr → kc given by the element v1 . Then, by Lemma 2.2, we are done. Similarly, we can show the injectivity of the other morphisms.
Proof of Lemma 3.2.1(contd.) (i) Combinig Claim (a) and (b), we get H|Si,C ⊆ OCr . So, deg H|Si,C ≤ 0. Similarly, deg
10
CHANDRANANDAN GANGOPADHYAY
H|Si,Pr−1 ≤ 0. ∗ (ii) Without loss of generality, assume j = 1. Then, H ⊆ π1,P r−1 Ω(1). Now, by claim (a), the compostion ∗ H|S1,Pr−1 → (F(r, d) ∩ π1,P → F(r, d) is injective outside a closed subset of r−1 Ω(1))|S 1,Pr−1 ∗ is injective outside a codimension ≥ 2. Hence, H|S1,Pr−1 → (F(r, d) ∩ π1,P r−1 Ω(1))|S 1,Pr−1 closed subset of codimension ≥ 2. Now, consider the map C × (C × Pr−1 )d C × C × (C × Pr−1 )d−1 given by π1 × π1,C × (π2,C × π2,Pr−1 ) × ...(πd,C × πd,Pr−1 ). Let V ⊆ C×C×(C×Pr−1 )d−1 is the open set with points of the form (c, c1 , (c2 , v2 ), ..(cd , vd )) with all c, ci ’s distinct. Then, over the inverse image of V with respect to the above map, ∗ i.e. the open set Pr−1 ×V , F(r, d)|Pr−1 ×V = OPr r−1 ×V . Hence, (F(r, d)∩πi,P r−1 Ω(1))|Pr−1 ×V = ∗ ∗ π1,Pr−1 Ω(1)|Pr−1 ×V . Therefore, with respect to points in V , F(r, d) ∩ π1,Pr−1 Ω(1))|S1,Pr−1 ∼ = Ω(1). Therefore, H|S1,Pr−1 → Ω(1) is injective outside a codimension 2 subset, and hence, deg H|S1,Pr−1 ≤ −1. ∗ (iii) Without loss of generality, we can assume that j = d. Then, since H πd,P r−1 Ω(1), hence, using the exact sequence (2.1) we have a non-zero map H → I0,d,C (1). Now, consider the map : C × (C × Pr−1 )d C × (C × Pr−1 )d−1 × Pr−1 given by π1 × (π1,C × π1,Pr−1 ) × ...(πd−1,C × πd−1,Pr−1 ) × πd,Pr−1 . By Lemma 3.2.3, for a generic choice of (c, (c1 , v1 ), .., (cd−1 , vd−1 ), vd ), the above non-zerp map H → I0,d,C (1) when restricted to Sd,C is non-zero, i.e. we have a non-zero map H|d,C → Ic , where the map is given by the composition H|d,C → OCr → OC , where the map OCr → OC is given by vd . Also, combining this with claim (a) and (b), we also get that H|Sd,C → OCr is injective, in particular, the kernel of the map H|d,C → Ic is also contained in Or . Hence, deg H|d,C ≤ deg Ic = −1. Similarly, we can show that deg H|0,C ≤ −1. Lemma 3.2.2. Let Y and Z are two smooth varieties over k. Let F be a torsion-free coherent sheaf on Y × Z and let G ⊆ F be a subsheaf such that F/V is also torsion free. Then, for a generic y ∈ Y , G|{y}×Z → F |{y}×Z is injective outside a closed subset ⊆ {y} × Z of codimension ≥ 2.
STABILITY OF SHEAVES OVER QUOT SCHEMES
11
Proof. Let W ⊆ Y × Z be the closed subset outside whic F, G and F/G are locally free. Let U = (Y ×Z)\W . Since Y ×Z is smooth, so, codim W ≥ 2. Consider the composition of maps W ⊆ Y × Z → Y . Let this map be denoted by f . Let V be the non-empty open set of Y such that for all y ∈ Y , f −1 (y) is empty, or has dimension = dim W − dim Y . Suppose all the fibers of points of V is empty. Then, ({y} × Z) ⊆ U . Therefore, F |{y}×Z , G|{y}×Z , (F/G)|{y}×Z are vector bundles and since F |{y}×Z → (F/G)|{y}×Z is surjective, so, kernel of this map is also a vector bundle. But, there is a surjective morphism from G|{y}×Z to the kernel. Since, both are vector bundles of same rank, so, it is an isomorphism. Suppose all fibers of points of V has dimension =dim W −dim Y . Then, dim ({y} × Z)− dim ({y} × Z) ∩ W ≥ 2. Now, the same argument as above shows that G|{y}×Z → F |{y}×Z is injective outside ({y} × Z) ∩ W . Lemma 3.2.3. Let Y be a smooth variety over k. Let F be a coherent sheaf and L a line bundle over C × Y . Suppose there is a non-zero map f : F → L. Then, for a generic choice of y ∈ Y , f |C×{y} is non-zero. Proof. Since, f is non-zero, Image(f ) ⊆ L is non-zero and hence, torsion-free. So, Image(f )|C×{y} = 0 ∀y ∈ Y . Therefore, it is enough to prove that if f is an inclusion, then for a generic choice of y ∈ Y , f |C×{y} is an inclusion. Also, tensoring f by L−1 , we can assume L = OC×Y and F = IZ i.e. the ideal sheaf of a closed subscheme Z ⊆C ×Y. Suppose codim Z = 1. Then, since C × Y is smooth, so IZ is a line bundle. Now, consider the composition of maps g : Z ⊆ C × Y → Y . There exists a non-empty open set V ⊆ Y such that fiber of each point in V is empty or has dim = dim Z− dim Y = 0. If fiber over each point of V is empty, then (C × V ) ∩ Z is empty, so, IZ |C×V = OC×V . Hence, for each point v ∈ V , IZ |C×{v} = OC×{v} . In particular, it is injective. Now, suppose fiber over each point of V has dimension 0. Then for any v ∈ V , Z∩(C×{y}) is a divisor on C. So, ker(OC×{y} → OZ∩(C×{y}) ) is a line bundle on C, and there is a surjection from IZ |C×{y} to this kernel. Hence, it is an isomorphism. If codim Z ≥ 2, then the image of Z in Y is a proper closed subset of Y , so there exists a non-empty open set V in Y such that fiber over each point is empty. Then, following the same argument as above, we can see that for any v ∈ V , IZ |C×{y} = OC×{y} . Theorem 3.2.4. Suppose d ≥ 2. Then if (i) r(r − 1)ab + (d + r − rd)bc + (d − r)ac ≥ 0 (ii) da + (d + r − rd)b ≥ 0 then, F(r, d) is semistable. If each of inequalities are strict, then F(r, d) is stable. if d=1, then F(r, d) is semistable iff rab − bc − ac ≥ 0, and stable iff the inequality is strict. Proof. Let 0 = H ⊆ F (r, d) such that F(r, d)/H is torsion-free. Then, there are three cases that we need to consider: ∗ (a) H πi,P r−1 Ω(1) ∀1 ≤ i ≤ d.
12
CHANDRANANDAN GANGOPADHYAY
∗ (b) H ⊆ πi,P r−1 Ω(1) ∀1 ≤ i ≤ d. ∗ ∗ (c)H ⊆ πi,Pr−1 Ω(1) 1 ≤ i ≤ j and H πi,P r−1 Ω(1) j + 1 ≤ i ≤ d, for some 1 ≤ j ≤ d − 1. ∗ (a) If H πi,P r−1 Ω(1) ∀1 ≤ i ≤ d, then by lemma 2.2, we have
deg H ≤ (−dac − bc)
(3.9)
(−dac − bc) r−1
(3.10)
Hence, μ (H) ≤ Therefore,
−dbc − dac −dac − bc − ) r r−1 (3.11) c = ((r + d − rd)b + da) ≥ 0 r(r − 1) The last inequality is by hypothesis (ii) of the theorem in the case d ≥ 2. If d = 1, then the inequality is automatically satisfied. Hence, μ (F(r, d)) ≥ μ (H). μ (F(r, d)) − μ (H) ≥ (
(b)
d i=1
∗ πi,P r−1 Ω(1) has rank r − d if r ≥ d + 1 and 0 otherwise. So, this case can happen
only when r ≥ d + 1.Therefore, assume r ≥ d + 1. d ∗ πi,P So, if H ⊆ r−1 Ω(1), then rank H ≤ r − d. i=1
Now, using lemma 2.2, we have deg H ≤ −dab(r − 1)
(3.12)
−dab(r − 1) r−d
(3.13)
So, μ (H) ≤ Then,
−dbc − dca −dab(r − 1) − ) r r−d (3.14) d[r(r − 1)ab − (r − d)bc − (r − d)ac] = r(r − d) Now, if d = 1, then r(r − 1)ab − (r − 1)bc − (r − 1)ac ≥ 0 is exactly the hypothesis of the theorem. If d ≥ 2, then the hypothesis (i) of the theorem implies rab(r − 1) − (r − d)ac − (r − d)bc ≥ (dr − d − r)bc − (r − d)bc = (dr − 2r)bc ≥ 0. Hence μ (F(r, d)) ≥ μ (H). r ∗ (c)Just like in case (b), if r ≤ d, then 1 ≤ j ≤ r − 1, since πi,P r−1 Ω(1) = 0. If d ≤ r + 1, μ (F(r, d)) − μ (H) ≥ (
i=1
then 1 ≤ j ≤ d − 1.Also, rank H ≤ r − j. Then, deg H ≤ (−jab(r − 1) − bc − (d − j)ac) μ (H) ≤ Then,
(−jab(r − 1) − bc − (d − j)ac) r−j
(3.15)
STABILITY OF SHEAVES OVER QUOT SCHEMES
−dbc − dac (−jab(r − 1) − bc − (d − j)ac) − ) r r−j (r(r − 1)jab + (−rd + dj + r)bc + (dj − rj)ac) = r(r − j) (j(r(r − 1)ab + dbc + (d − r)ac) + (r − rd)bc) = r(r − j)
13
μ (F(r, d)) − μ (H) ≥ (
(3.16)
If d ≥ 2, by hypothesis (ii), we have r(r − 1)ab + dbc + (d − r)ac + (r − rd)bc ≥ 0. So, r(r − 1)ab + dbc + (d − r)ac ≥ bc(rd − r) ≥ 0. Hence, for any j ≥ 1, j(r(r − 1)ab + dbc + (d − r)ac) + (r − rd)bc ≥ 0. Hence, μ (F(r, d)) ≥ μ (H). (This case cannot happen for d = 1.) Hence, F(r, d) is semistable. Clearly, if each of the inequalities is strict, then F(r, d) is stable. ∗ For d = 1, the inequality rab−bc−ac ≥ 0 is equivalent to the inequality μ (π1,P r−1 Ω(1)) ≤ μ (F(r, 1))). This follows from case (b). So, it is also a necessary condition.
3.3 Semistability and Stability of A(r, d) For a, b > 0, let (πC∗ OC (P ))a C × Q(r, d). Then, we have
∗ πQ (det ((πQ )∗ (B(r, d)
πC∗ OC (lP )))b ) be ample on
Theorem 3.3. (i) Suppose d ≥ 2. Then, if da + (d + r − rd)bl ≥ 0, then with respect to the above ample line bundle, A(r, d) is semistable, and if the inequality is strict, it is stable. (ii) If d = 1, then if ral − bl − a ≥ 0, then A(r, d) is semistable. If the inequality is strict, then it is stable. Proof. (i) Using Corollary 2.3, it is enough to show that F(r, d) is semistable with respect to the polarisation (a, bl, b). By Theorem 3.2.4 , we get that if (a) r(r − 1)abl + (d + r − rd)b2 l + (d − r)ab ≥ 0 (b) da + (d + r − rd)bl ≥ 0 Then, F(r, d) is semistable. Now, it is easy to see that the inequality (b) implies (a). Similarly, all other cases follow from Theorem 3.2.4.
4. Stability of direct image of A(r, d) If det((πQ )∗ (B(r, d)
πC∗ OC (lP ))) is ample on Q(r, d), then
14
CHANDRANANDAN GANGOPADHYAY
Theorem 4.1. (πQ )∗ A(r, d) is stable with respect to the above polarisation on Q(r, d). Lemma 4.2 If r ≥ d + 1, then (πQ )∗ A(r, d) is a torsion-free sheaf of rank r − d. If r ≤ d, then (πQ )∗ A(r, d) = 0.
Proof. The rank of (πQ )∗ A(r, d) is minimum of dimension of H 0 (C × {q}, A(r, d)|C×{q} ) as q varies over Q(r, d). This follows from semi-continuity theorem. Now, for any q ∈ Q(r, d), we have the exact sequence: A(r, d)|C×{q}
0
r OC×{q}
B(r, d)|C×{q}
0
Using the long exact sequence of cohomology, we get: H 0 (C, A(r, d)|C×{q} )
0
kr
kd
Hence, r − d ≤ dim H 0 (C, A(r, d)|C×{q} ) ≤ r. Now, if r ≥ d + 1, then consider any surjection OCr → OCd and consider the compostion of d kc , for some point c ∈ C, where the map is given by restriction this map with OCd → i=1
of each copy of OC to kc . Then, such OCr →
d i=1
kc taking H 0 , we get a surjection
k r → k d . Hence, its kernel has dimension r − d. Therefore, minimum dimension of H 0 (C × {q}, A(r, d)|C×{q} ) as q varies over Q(r, d) is r − d. Hence, rank of A(r, d) is r − d. r d−r kc kcj , where c, cj are all distinct, and If r ≤ d, consider the morphism OCr → the map OCr →
r i=1
i=1
j=1
kc when restricted to c gives an ismorphism of vector spaces, and
1 ≤ j ≤ d − r, let OCr → kcj is given by any surjection. Then, taking global sections of r d−r OCr → kc kcj , we get an inclusion k r → k d . Hence, minimum rank of A(r, d)|C×{q} i=1
j=1
r is 0. Since, (πQ )∗ A(r, d) ⊆ OQ , so, this implies (πQ )∗ A(r, d) = 0.
Remark : From now on, we will assume r ≥ d + 2, since, if r ≤ d then (πQ )∗ A(r, d) = 0, and if r = d + 1, then it is a rank 1 sheaf.
Recall that we have the following diagram:
STABILITY OF SHEAVES OVER QUOT SCHEMES
C ×U
idC ×Φ
C × Q(r, d)
π2
U
15
πQ Φ
Q(r, d)
Lemma 4.3. The natural map Φ∗ (πQ )∗ A(r, d) → (π2 )∗ (idC × Φ)∗ A(r, d) is an isomorphism outside a closed subset of codimension ≥ 2.
Proof. For any permutation (i1 , i2 , .., id ) of (1, 2, .., d), define Zi1 ,i2 ,..,id = {u ∈ U|pid ,Pr−1 (u) ∈ subspace generated bypi1 ,Pr−1 (u), pi2 ,Pr−1 (u), .., pid−1 ,Pr−1 (u) in Pr−1 }. Since r ≥ d + 2, Zi1 ,i2 ,..,id is a closed subset of codim ≥ 2. Let U = U \ Zi1 ,i2 ,..,id .
i1 ,i2 ,..,id (r−1) : ... : vi ] be a choice of Let u = ((c1 , v1 ), (c2 , v2 ), .., (cd , vd )) ∈ U . Let vi = d r ∗ → πi,P homogeneous coordinate for vi . Then, the quotient OC×U r−1 O(1)|0,i,C on i=1 d kci , where OCr → kci is given C × U , when restricted to C × {u} becomes OCr → i=1 (0) (1) (r−1) . Hence, the map by the element vi ∈ Pr−1 ,i.e. e(0) → vi , e(1) → vi , .., e(r−1) → vi ⎡ (0) (1) (r−1) ⎤ v1 v1 ... v1 (0) (1) (r−1) ⎥ ⎢ v2 v2 ... v2 d ⎢ ⎥ ⎥. H 0 (C, OCr ) → H 0 (C, kci ) is given by the matrix ⎢ ⎢ . ⎥ i=1 ⎣ ⎦ (0) [vi
.
(0) vd
(1)
(r−1)
vd ... vd Since, by definition of U , the set {v1 , v2 , ..., vd } is linearly independent, i.e. the rows of the above matrix are linearly independent. Therefore, rank of the above matrix is d and d hence, the morphism H 0 (C, OCr ) → H 0 (C, kci ) is surjective. i=1
Now, by Grauert’s theorem [3, Chapter III,Corollary 12.9 ], we have the following commutative diagram:
r )|u (π2 )∗ (OC×U
(π2 )∗ (
d i=1
∗ πi,P r−1 O(1)|0,i,C )|u ∼ =
∼ =
H 0 (C, OCr )
H 0 (C,
d i=1
r Therefore, the map (π2 )∗ (OC×U )|u → (π2 )∗ (
we have the following commutative diagram:
k ci )
d i=1
∗ πi,P r−1 O(1)|0,i,C )|u is surjective. Now,
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CHANDRANANDAN GANGOPADHYAY
(π2 )∗ F(r, d)|u
r (π2 )∗ (OC×U )|u
(π2 )∗ (
d i=1
∗ πi,P r−1 O(1)|0,i,C )|u ∼ =
∼ =
0
H 0 (C, F(r, d)|C×{u} )
0
r H 0 (C, OC×{u} )
H 0 (C,
d i=1
k ci )
0
Here, the rows are exact and hence, the map (π2 )∗ F(r, d)|u → H 0 (C, F(r, d)|C×{u} ) is surjective. Then, using [3, Chapter III, Theorem 12.11] , it is an isomorphism. d r r r → B(r, d)|C×{Φ(u)} = OC×{u} → kci . So,H 0 (C, OC×{Φ(u)} )→ Now, for Φ(u), the map OC×{Φ(u)} i=1
H 0 (C, B(r, d)|C×{Φ(u)} )) is again surjective. Therefore, applying the same argument as above, we get that (πQ )∗ A(r, d)|Φ(u) ∼ = H 0 (C, A(r, d)|C×{Φ(u)} ). Therefore, we have: Φ∗ (πQ )∗ A(r, d)|u = (πQ )∗ A(r, d)|Φ(u)
(π2 )∗ F(r, d)|u ∼ =
∼ =
H 0 (C, F(r, d)|C×{u} ) Hence, ∀ u ∈ U , Φ∗ (πQ )∗ A(r, d)|u → (π2 )∗ (idC ×Φ)∗ A(r, d)|u is an isomorphism. There fore, Φ∗ (πQ )∗ A(r, d)|U → (π2 )∗ (idC × Φ)∗ A(r, d)|U is an isomorphism.
Corollary 4.4. If (π2 )∗ F(r, d) is stable with respect to the polarisation (p∗i,C OC (l) p∗i,Pr−1 OPr−1 (1)), then (πQ )∗ A(r, d) is stable with respect to i=1 det(π∗ (B(r, d) πC∗ OC (l))). d
Proof. This follows from Lemma 1.1, Theorem 2.1(iii) and Lemma 4.3.
Proof of theorem 4.1. By the above corollary, it is enough to show that (π2 )∗ F(r, d) d over X = (C × Pr−1 )d is stable with respect to (p∗i,C OC (b) p∗i,Pr−1 OPr−1 (c)), for any b, c > 0. Now, we have the following left exact sequence:
0
(π2 )∗ F(r, d)
OXr
i=1
(π2 )∗ (
d i=1
∗ πi,P r−1 O(1)|0,i,C )
STABILITY OF SHEAVES OVER QUOT SCHEMES
Claim: (π2 )∗ ( OXr → (π2 )∗ (
d i=1
d i=1
17
∗ ∼ p∗ r−1 O(1) and the map πi,P r−1 O(1)|0,i,C ) = i,P d
i=1
d ∗ ∼ p∗ r−1 O(1) is such that the projection to i-th facπi,P r−1 O(1)|0,i,C ) = i,P i=1
tor, i.e. the map OXr → p∗i,Pr−1 O(1) is given by p∗i,Pr−1 (OPr r−1 → O(1)). proof of claim. Now, the map d d ∗ r ∗ (π2 )∗ (πi,P OXr → (π2 )∗ ( πi,P r−1 O(1)|0,i,C ) = OX → r−1 O(1)|0,i,C ). i=1
i=1
∗ ∼ ∗ So, it is enough to show (π2 )∗ π1,P r−1 )O(1)|0,i,C = p1,Pr−1 O(1) and r ∗ ∗ r (π2 )∗ (OC×X → π1,P r−1 O(1)|0,1,C ) = p1,Pr−1 (OPr−1 → O(1)).
Now, 0,1,C = X and if, j :0,1,C → C ×X is the inclusion, then, π2 ◦j is the isomorphism and π1,Pr−1 ◦ j = p1,Pr−1 . ∗ ∗ ∗ ∗ So, (π2 )∗ (π1,P r−1 O(1)|1,i,C ) = (π2 )∗ j∗ (j πi,Pr−1 O(1)) = pi,Pr−1 O(1). r ∗ Also, the map OC×X → π1,P r−1 O(1)|0,1,C factors as : ∗ π1,P r−1 O(1)|0,1,C
Or 0,1,C
r OC×X
where the second map is just j∗ (OXr → p∗1,Pr−1 O(1)), so, (π2 )∗ (j∗ )(OXr → p∗1,Pr−1 O(1)) gives r the required morphism. The only thing we need to show is that (π2 )∗ (OC×X → Or 0,1,C ) is r the identity map from OXr → OXr . Now, (π2 )∗ (Or 0,1,C ) = OXr , and H 0 (C × X , OC×X )→ 0 r H (C × X , O0,1,C ) is identity. Hence, the result follows. Proof of theorem 4.1(contd.). Using the claim, we get that d p∗i,Pr−1 Ω(1) ⊆ OXr . (π2 )∗ F(r, d) = i=1
Now, by the proof of the Lemma 4.2, we get that the following sequence is exact outside a codimension ≥ 2 subset:
(π2 )∗ F(r, d)
0
d
OXr
i=1
p∗i,Pr−1 O(1)
Therefore, using Corollary 3.1.2, we get: d (rd − 1)!(bd )(cr−1 )d−1 cr−2 (−d). deg (π2 )∗ F(r, d) = − deg p∗i,Pr−1 O(1) = ((r − 1)!)d−1 (r − 2)! i=1 d p∗i,Pr−1 Ω(1) such that the quotient is torsion free. Then, using Lemma Now, let H ⊆ i=1
3.2.2, we have H|S1,Pr−1 → ( of Pr−1 . Also, note that
d i=1
d i=1
p∗i,Pr−1 Ω(1))|Si,Pr−1 outside a codimension ≥ 2 closed subset
p∗i,Pr−1 Ω(1) ⊆
d−1 i=1
p∗i,Pr−1 Ω(1) is such that the quotient is torsion
18
CHANDRANANDAN GANGOPADHYAY
free. This follows from the following exact sequence: d
0
i=1
Similarly, we have H ⊆
d−1
p∗i,Pr−1 Ω(1)
d i=1
i=2
p∗i,Pr−1 Ω(1) ⊆
d−1 i=1
p∗i,Pr−1 Ω(1)
p∗i,Pr−1 Ω(1) ⊆
p∗d,Pr−1 O(1)
d−2 i=1
(4.1)
p∗i,Pr−1 Ω(1) ⊆ ... ⊆ p∗1,Pr−1 Ω(1)
such that each successive quotient is torsion free. Hence, H|S1,Pr−1 → p∗1,Pr−1 Ω(1) is injective outside a codimension ≥ 2 closed subset of Pr−1 |S1,Pr−1 . So, deg H|S1,Pr−1 ≤ −1. Same is true for any i. Therefore, using Corollary 3.1.2, we get that deg H ≤ d d (rd − 1)!bd (cr−1 )d−1 cr−2 ∗ (−d) = deg p Ω(1).Since, rank H < rank p∗i,Pr−1 Ω(1), r−1 i,P ((r − 1)!)d−1 i=1 i=1 d d p∗i,Pr−1 Ω(1) is stable. hence, μ(H) < μ( p∗i,Pr−1 Ω(1)). Hence, i=1
i=1
5. Stability of sheaves over flag schemes Fix d = (d1 , d2 , .., dk ) ∈ Nk with d1 d2 .. dk 0 and let r ≥ 1. Let Q(r, d) be scheme whose closed points correspond to filtered quotients OCr → B1 → B2 → .. → Bk such that each OCr → Bj is surjective and Bj is a torsion sheaf of length dj for 1 ≤ j ≤ k. Then, over Q(r, d) we have chain of quotients: r OC×Q(r,d)
B1 (r, d)
B2 (r, d)
...
Bk (r, d)
r Let us denote the kernel of OC×Q(r,d) → Bj (r, d) by Aj (r, d). Let πC and πQ be the projections from C × Q(r, d) to C and Q(r, d) respectively. k Then, det (πQ )∗ (Bj (r, d) π ∗ OC (lj )) is ample for lj >> 0 ∀j. Let us fix such an j=1
ample line bundle and let it be denoted by L. Then, we have the following theorem: Theorem 5.1. (πQ )∗ Ai (r, d) is stable with respect to L. Proof. Let d1 = d. Following the notation of the previous sections, we have X = (C × d r ∗ Pr−1 ). Recall that over C × X we have the morphism q : OC×X → πi,P r−1 O(1)|0,i,C . i=1
Now, consider the chain of morphisms over C × U , r OC×X
q
→ −
d i=1
∗ πi,P r−1 O(1)|0,i,C
→
d2 i=1
∗ πi,P r−1 O(1)|0,i,C
→ ... →
dk i=1
∗ πi,P r−1 O(1)|0,i,C
(5.1)
STABILITY OF SHEAVES OVER QUOT SCHEMES
where
dj i=1
∗ πi,P r−1 O(1)|0,i,C →
d j+1 i=1
19
∗ πi,P r−1 O(1)|0,i,C is just the projection to the first dj+1
factors. Now, by theorem 2.1, q|C×U is a surjection, hence, each of the morphisms dj r ∗ OC×X → πi,P r−1 O(1)|0,i,C in the above chain, when restricted to C × U is surjeci=1
tive. Therefore, by universal property of Q(r, d) we have a map Φd : U → Q(r, d) such that dj r r ∗ → Bj (r, d)) = (OC×X → πi,P (i) Φ∗d (OC×Q(r,d) r−1 O(1)|0,i,C )|C×U i=1
r (ii)(idC × Φd )∗ A(r, d) = ker (OC×X →
dj
i=1
∗ πi,P r−1 O(1)|0,i,C )|C×U
d (iii)(idC × Φd )∗ L ∼ = ( (p∗i,C O(ai ) p∗i,Pr−1 OPr−1 (bi )))|U for some ai , bi > 0. i=1
(iv)(a) If dj = r + 1, then rank of (πQ )∗ Aj (r, d) is 1. If dj ≤ r, then (πQ )∗ Aj (r, d) = 0. (b)If dj ≤ r + 2, then (Φd )∗ (πQ )∗ Aj (r, d) ∼ = (π2 )∗ (idC × Φd )∗ Aj (r, d) outside a closed subset of X of codimension ≥ 2. The proofs of (i), (ii) and (iii) are same as in Theorem 2.1. The proof of (iv)(a) is same as in Lemma 4.2 and the proof of (iv)(b) is same as in Lemma 4.3. By (iv)(a), if r ≤ d + 1,then we are done. So, let r ≥ dj + 2. dj r ∗ πi,P Let us denote ker (OC×X → r−1 O(1)|0,i,C ) by Fj (r, d). i=1
Then, using Lemma 1.1 , (iii) and (iv)(b) it is enough to show that (π2 )∗ Fj (r, d) is stable d with respect to the polarisation (p∗i,C O(ai ) p∗i,Pr−1 OPr−1 (bi )) for any ai , bi > 0. i=1
Now, by the claim in the proof of Theorem 4.1. , (π2 )∗ Fj (r, d) = rest of the proof is same as in Theorem 4.1.
dj i=1
p∗i,Pr−1 Ω(1). Now, the
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CHANDRANANDAN GANGOPADHYAY
References [1] [2] [3]
D. Huybrechts and M. Lehn, The Geometry of Moduli Spaces of Sheaves, Cambridge University Press, Second Edition(2010). V. Balaji, L. Brambila-Paz and P. E. Newstead, Stability of the Poincare bundle, Math. Nachr. 188 (1997), 5–15. R. Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, (1977).
School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India E-mail address: [email protected]