COMMUTING MATRICES AND THE HILBERT SCHEME OF POINTS

arXiv:1304.3028v3 [math.AG] 20 Aug 2013

COMMUTING MATRICES AND THE HILBERT SCHEME OF POINTS ON AFFINE SPACES P. BORGES DOS SANTOS, ABDELMOUBINE A. HENNI, AND MARCOS JARDIM Abstract. We give a linear algebraic and a monadic descriptions of the Hilbert scheme of points on the affine space of dimension n which naturally extends Nakajima’s representation of the Hilbert scheme of points on the plane. As an application of our ideas and recent results from the literature on commuting matrices, we show that the Hilbert scheme of c points on (C3 ) is irreducible for c ≤ 10.

Contents 1. Introduction 2. Commuting matrices and stable ADHM data 3. Parametrization of the Hilbert scheme of c Points in Cn 4. Extended monads and perfect extended monads 4.1. l−extended monads 4.2. Perfect extended monads 5. Ideal sheaves of zero-dimensional subschemes of Pn 6. The P3 case 7. Representability of the Hilbert functor of points 8. The Hilbert–Chow map 9. The Hilbert scheme of points on affine varieties 9.1. Scheme structure on MY (c) [c] 9.2. The schematic isomorphism MY (c) ∼ = HilbY 10. Irreducible components of the Hilbert scheme of points References

1 3 6 10 10 13 15 16 22 24 25 26 27 29 30

1. Introduction [c]

The Hilbert scheme Hilb (Cn ) of c points in the affine space of dimension n parametrizes 0-dimensional subschemes of Cd of length c. The case of n = 2 is much studied, though less is known about the higher dimensional cases. The linear algebraic and monadic descriptions of the n = 2 case given by Nakajima in [16, Chapters 1 & 2] is particularly relevant to us. One of the goals of this paper is to give analogous descritions of the Hilbert scheme Hilb[c] (Cn ) of c points on Cn , naturally extending Nakajima’s representation of the Hilb[c] (C2 ). This goal is attained in the first part of the paper, Sections 2 through 7. More precisely, let V and W be complex vector spaces of dimension c and 1, respectively. Let B1 , . . . , Bn be operators on V commuting with each other and 1

2

P. BORGES DOS SANTOS, ABDELMOUBINE A. HENNI, AND MARCOS JARDIM

consider a map I : W → V . The (n + 1)-tuple (B1 , . . . , Bn , I) is said to be stable if there is no proper subspace S ⊂ V which is invariant under each operator Bk and contains the image of I. The group GL(V ) acts on the set of all such (n + 1)-tuple by change of basis on V . We prove that there is a one-to-one corresponding between the following objects: (1) ideals J in the ring of polynomials C[x1 , . . . , xn ] whose quotient has dimension c; (2) stable (n + 1)-tuple (B1 , . . . , Bn , I) with dim V = c, modulo the action of GL(V ); (3) complexes of the form, called perfect extended monads: α1−n

V1−n ⊗ OPn (1 − n)

/ V2−n ⊗ OPn (2 − n)

...

α−1

/ V0 ⊗ OPn

α0

/ V1 ⊗ OPn (1)

n where V1 := V , V0 = V ⊕n ⊕ C and Vi = V ⊕(1−i) for i < 0, which are exact everywhere except at degree 0 (grading of the complex is given by the twisting).

Furthermore, using the above correspondence, we also show that the Hilb[c] (Cn ) is isomorphic (as a scheme) to a GIT quotient of C(n, c) × Hom(W, V ) by GL(V ), where C(n, c) denotes the variety of n commuting c × c matrices. The correspondence between items (1) and (2) as well as the isomorphism between Hilbert scheme and the GIT quotient of C(n, c) × Hom(W, V ) by GL(V ) are already present in the representation theory literature, see for instance [19], [20, Appendix by M. V. Nori] and [22], and more recently [6, 9, 23]. However, our presentation is more down-to-earth and closer to Nakajima’s description which is familiar to algebraic geometers. The correspondence with the so-called perfect extended monads is new. In fact, we introduce in Section 4 a new class of objects, extended monads (cf. Definition 4.1), which generalize the usual monads originally introduced by Horrocks in the 1960’s [11] and much studied by several authors since then. The basic theory of extended monads is developed here, with a focus on what we call perfect extended monads. We provide a cohomological characterization of the sheaves that arise as cohomology of a perfect extended monad on projective spaces (see Proposition 4.8 below), showing, in particular, that ideal sheaves of zero dimensional subschemes do satisfy the required conditions. We complement our discussion on the parametrization via linear algebra of the Hilbert scheme of points on affine varieties in two ways. First, in Section 8, we provide a desciption of the Hilbert–Chow morphism from Hilb[c] (Cn ) to the symmetric product of c copies of Cn in terms of our linear data. Second, we provide a linear algebraic description of Hilbert scheme Hilb[c] (Y) of c points on an affine variety Y ⊂ Cn . More precisely, suppose that Y is given by algebraic equations f1 = · · · = fl = 0; we show in Section 9 that a point in Hilb[c] (Y) corresponds to a stable (n + 1)-tuple (B1 , . . . , Bn , I) such that fk (B1 , . . . , Bn ) = 0 for each k = 1, . . . , l. Morever, such correspondence yields a schematic isomorphism between Hilb[c] (Y) and the variety of commuting matrices satisfying fk (B1 , . . . , Bn ) = 0 plus a vector, modulo GL(V ). Finally, as an application of our ideas, it is not difficult to see that Hilb[c] (Cn ) is irreducible whenever C(n, c) is irreducible, see details in Section 10 below. It then follows from recent results due to Sivic [21, Theorems 26 & 32], that Hilb[c] (C3 ) is

COMMUTING MATRICES AND THE HILBERT SCHEME OF POINTS

3

irreducible if c ≤ 10, while this was known to be the case only for c ≤ 8, cf. [2, Theorem 1.1]. Acknowledgments. We would like to thank D. Erman for useful discussions and suggestions about the irreducibility results of Hilbert schemes of points. PBS research was supported by CNPq and the FAPESP PhD grant number 2009/16646-1. AAH is supported by the FAPESP post-doctoral grant number 2009/12576-9. MJ is partially supported by the CNPq grant number 302477/20101 and the FAPESP grant number 2011/01071-3. 2. Commuting matrices and stable ADHM data In this section we shall introduce the necessary material to our construction: let V be a complex vector space of dimension c and let B0 , B1 , . . . , Bn−1 ∈ End(V ) be n linear operators on V . Definition 2.1. The variety C(n, c) of n commuting linear operators on V is the subvariety of End(V )⊕n whose points are the set of n-tuples (B0 , B1 , . . . , Bn−1 ) that commutes two by two, that is, C(n, c) = (B0 , B1 , . . . , Bn−1 ) ∈ End(V )⊕n | [Bi , Bj ] = 0, ∀i 6= j The commutation relations can be thought of as a system of n2 c2 homogeneous equations of degree 2 in nc2 variables. Let W be a 1-dimensional complex vector space; one can form the space B := End(V )⊕n ⊕ Hom(W, V ) whose points are represented by the (n + 1)-tuple X = (B0 , B1 , . . . , Bn−1 , I) that will be called an ADHM datum. We then define the variety of ADHM data V(n, c) as the subvariety of B given by V(n, c) := C(n, c) × Hom(W, V ). Definition 2.2. An ADHM datum X = (B0 , B1 , . . . , Bn−1 , I) ∈ B is said to be stable if there is no proper subspace S ( V such that B0 (S), B1 (S), · · · , Bn−1 (S), I(W ) ⊂ S. The set of stable points in B will be denoted by Bst ; V(n, c)st := Bst ∩ V(n, c) will denote the set of stable points in V(n, c). Definition 2.3. The stabilizing subspace ΣX of an ADHM datum X ∈ B is the intersection of all subspaces S ⊂ V such that B0 (S), B1 (S), · · · , Bn−1 (S), I(W ) ⊂ S. It is easy to see that X is stable if and only if ΣX = V . The restricted ADHM datum X|ΣX = (B0 |ΣX , B1 |ΣX , · · · , Bn−1 |ΣX , I|ΣX ) is stable in B|ΣX = End(ΣX )⊕n ⊕ Hom(W, ΣX ). The space ΣX ⊂ V is the smallest subspace which makes the datum X|ΣX stable, hence the name. For each ADHM datum X = (B0 , . . . , Bn−1 , I) ∈ B, we consider the linear map n Rn (X) : W ⊕c −→ V defined by W ⊕c

Rn (X) : c−1 M

k0 ,...,kn−1 =0

n

wk0 ,...,kn−1

−→ 7−→

V c−1 X

k0 ,...,kn−1 =0

k

n−1 Iwk0 ,...,kn−1 B0k0 B1k1 . . . Bn−1

4

P. BORGES DOS SANTOS, ABDELMOUBINE A. HENNI, AND MARCOS JARDIM n

One might think of Rn as a regular morphism B → Hom(W ⊕c , V ), hence continuous in the Zariski topology. Proposition 2.4. For every X ∈ B one has (1) Im Rn (X) ⊆ ΣX ; (2) if Rn (X) is surjective, then X is stable. Proof. For any S ⊆ V satisfying B0 (S), B1 (S), . . . , Bn−1 (S), I(W ) ⊂ S we have Im Rn (X) ⊆ S which in particular implies our first assertion. Moreover, if Rn (X) is surjective then we have c = rk Rn (X) ≤ dim ΣX ≤ c. Therefore dim ΣX = c and X is stable. If the ADHM datum X is in V(n, c), then we obtain the following stronger characterization. Proposition 2.5. For every datum X = (B0 , . . . , Bn−1 , I) ∈ V(n, c) one has: (1) Im Rn (X) = ΣX . (2) Rn (X) is surjective if and only if X is stable. Proof. Note first that the second claim follows easily from Proposition 2.4 and the first claim. To prove the first claim, we only need to prove the inverse inclusion ΣX ⊆ Im Rn (X) for those ADHM data which belong to V(n, c) since the inclusion Im Rn (X) ⊆ ΣX holds for all X ∈ B. For this end, it is enough show that Im Rn (X) is Bi -invariant, for all i ∈ {0, . . . , n − 1}, and I(W ) ⊆ Im Rn (X). It is easy to see that I(W ) ⊆ Im Rn (X), so the results follows by showing that Im Rn (X) is Bi -invariant, for all i ∈ {0, . . . , n − 1}; let c−1 X kn−1 Iwk0 ,...,kn−1 ∈ Im Rn (X). B0k0 B1k1 . . . Bn−1 k0 ,...,kn−1 =0

For X ∈ V(n, c) one has the following identity 

Bi 

c−1 X

k0 ,...,kn−1 =0



kn−1 B0k0 B1k1 . . . Bn−1 Iwk0 ,...,kn−1  c−1 X

= k

n−1 Bi B0k0 B1k1 . . . Bn−1 Iwk0 ,...,kn−1

k0 ,...,kn−1 =0

=

c−1 X

k

n−1 Iwk0 ,...,kn−1 B0k0 . . .Bic . . . Bn−1

k0 ,...,b ki ,...,kn−1 =0

+

c−1 X

c−1 X

k

n−1 Iwk0 ,...,kn−1 . B0k0 . . . Biki . . . Bn−1

k0 ,...,b ki ,...,k0 =0 ki =1

By the symbol ˆ we mean omitting the term bellow it from the expression. It is clear that the second factor of the sum, in the lower line of the expression above, belongs to Im Rn (X). To see that also the first factor belongs to Im Rn (X), notice that the characteristic polynomial of Bi is of the form p(x) = xc + ac−1 xc−1 + . . . +


5

a1 x+a0 . Hence, by Cayley-Hamilton Theorem, it follows that Bic is given by a linear combination of lower powers of Bi , i. e., Bic = −(ac−1 Bic−1 + . . . + a1 Bi + a0 IV ). Pc−1 kn−1 B k0 . . . Bic . . . Bn−1 With this, we conclude that Iwk0 ,...,kn−1 ∈ k0 ,...,b ki ,...,kn−1 =0 0 Im Rn (X) which, in particular means, that Im Rn (X) is Bi -invariant. Finally, since ΣX is the smallest subspace of V with this properties, it then follows that ΣX ⊆ Im Rn (X). Next, we introduce the action of the linear group G := GL(V ) on B. For all g ∈ G and X = (B0 , . . . , Bn−1 , I) ∈ B, this action is given by g · (B0 , . . . , Bn−1 , I) = (gB0 g −1 , . . . , gBn−1 g −1 , gI). For a fixed ADHM datum X, we will denote by GX its stabilizer subgroup: GX := {g ∈ G | gX = X} ⊆ G. It is easy to see that X is stable if and only if gX is stable, and that G acts on V(n, c). We conclude this section with two results relating stability in the sense of Definition 2.2 with GIT stability. Proposition 2.6. If X ∈ V(n, c)st , then its stabilizer subgroup GX is trivial. Proof. Let X = (B0 , . . . , Bn−0 , I) be a stable ADHM datum and suppose that there exists an element g 6= 1 in G such that gI = I and gBi g −1 = Bi for all i ∈ {0, . . . , n − 1}. Then ker(g − 1) is Bi -invariant, for all i ∈ {0, . . . , n − 1}, and Im I ⊆ ker(g − 1). Since X is stable, then ker(g − 1) ⊂ V must be the trivial subspace. Hence g must be the identity. Let Γ(V(n, c)) be the ring of regular functions on V(n, c). Fix l > 0, and consider the group homomorphism χ : G → C∗ given by χ(g) = (det g)l . This can be used for the of construction a suitable linearization of the G−action on V(n, c), that is, to lift the action of G on V(n, c) to an action on V(n, c) × C as follows: g · (X, z) := (g · X, χ(g)−1 z) for any ADHM datum X ∈ V(n, c) and z ∈ C. Then one can form the scheme   M i V(n, c)//χ G := Proj  Γ(V(n, c))G,χ  i≥0

where

i Γ(V(n, c))G,χ := f ∈ Γ(V(n, c)) | f (g · X) = χ(g)−1 · f (X), ∀g ∈ G .

The scheme V(n, c)//χ G is projective over the ring Γ(V(n, c))G and quasi-projective over C. Proposition 2.7. The orbit G·(X, z) is closed, for z 6= 0, if and only if the ADHM datum X ∈ V(n, c) is a stable. Proof. First, suppose that the orbit G · (X, z) is not closed, then there is a 1-parameter subgroup λ : C∗ → G such that the limit (L, w) := limt→0 λ(t) · (X, z) exists but does not belong to the orbit G · (X, z). The existence of the limit (L, w) implies that det(λ(t)) = tN for some N ≤ 0. If N = 0 then λ(t) = IV , which contradicts the fact that the limit does not belong to the orbit G · (X, z). Thus

6


L N < 0. Now, take the weight decomposition V = m V (m) of the space V, with respect to λ. Then the existence of a limit implies that M M B0 (V (m)), B1 (V (m)), . . . , Bn−1 (V (m)) ⊂ V (m) and I(W ) ⊂ V (m). n≥m

n≥0

L

The space S := n≥0 V (m) ∈ V is proper since N < 0. Moreover, one has B0 (S), B1 (S), . . . , Bn−1 (S) ⊂ S and I(W ) ⊂ S. Hence X is not stable. Conversely, suppose that the ADHM datum X = (B0 , B1 , . . . , Bn−1 , I) is not stable. Then there exists a subspace S ⊂ V such that B0 (S), B1 (S), . . . , Bn−1 (S) ⊂ S and I(W ) ⊂ S. Let T ⊂ V be a subspace such that V = S ⊕ T. With respect to this decomposition, one can write the linear maps Bi , for 0 ≥ i ≥ n − 1 and I, as follows ⋆ ⋆ ⋆ i B = , for 0 ≥ i ≥ n − 1 I = . 0 ⋆ 0 Now, define the 1-parameter subgroup as IS λ(t) = 0 then −1

λ(t)Bi λ(t)

=

⋆ 0

0 t−1 IT t⋆ ⋆

,

and λ(t)I = I.

It follows that the limit L = limt→0 λ(t) · X exists and limt→0 λ(t) · (X, z) = (L, 0), which means that the orbit is closed within V(n, c) × C∗ From Propositions 2.6 and 2.7 and since the group G is reductive, it follows that the quotient space M(n, c) := V(n, c)//χ G is a good categorical quotient [15, Thm. 1.10]. Furthermore, GIT tells us that the GIT quotient M(n, c) is the space of orbits G · X ⊂ V(n, c) such that the lifted orbit G · (X, z) is closed within V(n, c) × C for all z 6= 0. We conclude therefore, from Proposition 2.7, that M(n, c) = V(n, c)st /G. 3. Parametrization of the Hilbert scheme of c Points in Cn As a set, the Hilbert scheme of c points on Cn is given by: Hilb[c] (Cn ) = {I ⊳ C[z0 , . . . , zn−1 ] | dimC (C[z0 , . . . , zn−1 ]/I) = c}. The existence of its schematic structure is a special case of the general result of Grothendieck [8]. Another explicit construction of the Hilbert scheme of points on the affine plane is given by Nakajima [16]. The reader may also consult [17] for more general results and examples. The aim of this section is to prove the following result Theorem 3.1. There exists a set-theoretical bijection between the quotient space M(n, c) and the Hilbert scheme of c points in Cn . We remark that this result will be strengthen, in Section 7 below, to a scheme theoretic isomorphism rather than just a bijective correspondence. Before proving the above result it will be useful to, first, establish a few lemmata.


7

Lemma 3.2. If X = (B0 , . . . , Bn−1 , I) ∈ V(n, c)st is a stable ADHM datum, then the map: ΦX : C [Z0 , . . . , Zn−1 ] −→ V p(Z0 , . . . , Zn−1 ) 7−→ p(B0 , . . . , Bn−1 )I(1) is a surjective linear transformation. In particular, C [Z0 , . . . , Zn−1 ] / ker ΦX is isomorphic to V . We Remark that the linear map ΦX is well-defined, since [Bi , Bj ] = 0, for all 0 ≤ i < j ≤ n − 1. Proof. Observe that Im I ⊆ Im ΦX since the elements of Im I consist of vectors of the form αI(1), for some constant α in C. The inverse image of such an element is simply the constant polynomial α itself. Moreover, the image Im ΦX of the map ΦX is Bi -invariant, for all 0 ≤ i ≤ n − 1, since all the Bi ’s commute. By stability of the ADHM datum X we must have Im ΦX = V, and hence ΦX is surjective. It is clear that ker ΦX ⊂ C [Z0 , . . . , Zn−1 ] is an ideal. Now, given any two polynomials p(Z0 , . . . , Zn−1 ) ∈ C [Z0 , . . . , Zn−1 ] and q(Z0 , . . . , Zn−1 ) ∈ ker ΦX , one has ΦX (p(Z0 , . . . , Zn−1 )q(Z0 , . . . , Zn−1 )) = p(B0 , . . . , Bn−1 )q(B0 , . . . , Bn−1 )I(1) = 0. Hence the isomorphism C [Z0 , . . . , Zn−1 ] / ker ΦX ≃ V. Let π : V(n, c)st → M(n, c) be the natural projection onto the orbit space M(n, c) and denote by [X] = [(B0 , . . . , Bn−1 , I)] the class of the ADHM datum X = (B0 , . . . , Bn−1 , I) in V(n, c)st , that is, [X] = π(X). Lemma 3.3. Let X, Y ∈ V(n, c)st be two stable ADHM data such that [X] = [Y ]. Then ker ΦX ≃ ker ΦY . Proof. Suppose that [X] = [(B0 , . . . , Bn−1 , I)] = [Y ] = [(A0 , . . . , An−1 , J)], then there exists an element g ∈ GL(V ) such that Ai = gBi g −1 , for all 0 ≤ i ≤ n − 1 and J = gI. Now, for any polynomial f ∈ C[Z0 , . . . , Zn−1 ] one has f (A0 , . . . , An−1 ) = gf (B0 , . . . , Bn−1 )g −1 , hence f (A0 , . . . , An−1 )J(1) = gf (B0 , . . . , Bn−1 )g −1 (gI(1)) = gf (B0 , . . . , Bn−1 )I(1), in other words, ΦY = gΦX . Since g is invertible, it then follows that ker ΦX ≃ ker ΦY . Lemma 3.4. nThe ADHM datum X = (B0 , . . . , Bn−1 , I) ∈ oV(n, c) is stable if and in−1 only if the set B0i0 · B1i1 · · · Bn−1 I(1) ∈ V | ik = 0, . . . , c − 1 spans V as a complex vector space. Proof. The result follows from item (2) of Proposition 2.5 and the fact that the set n o in−1 i0 i1 B0 · B1 · · · Bn−1 I(1) ∈ V | ik = 0, . . . , c − 1 spans Im Rn (X). We are finally in position to complete the Proof of Theorem 3.1. Proof of Theorem 3.1. We will consider the map Ψ : M(n, c) −→ Hilb[c] (Cn ) . [X] 7−→ ker ΦX

8


which associates the ideal ker ΦX to the class [X] = [(B0 , . . . , Bn−1 , I)] of a stable st ADHM datum X = (B0 , . . . , Bn−1 , I) ∈ V(n, c) . By Lemma 3.3, the map Ψ is well-defined and it is clear from lemma 3.2 that ker ΦX belong to Hilb[c] (Cn ). Inversely, we define the map Ψ′ :

Hilb[c] (Cn ) −→ M(n, c) J 7−→ [(B0 , . . . , Bn−1 , I)]

[c]

from HilbCn to M(n, c) as the following: Given an ideal J ∈ Hilb[c] (Cn ) we denote by V = C [Z0 , . . . , Zn−1 ] /J the vector space associated to it. The multiplication by Zi mod J define endomorphisms Bi ∈ End(V ), for 0 ≤ i ≤ n − 1, in the following way Bi :

(1)

V [p(Z0 , . . . , Zn−1 )]

−→ V 7−→ [Zi p(Z0 , . . . , Zn−1 )]

One can also define I ∈ Hom(W, V ) as the linear mapping which associates to the unit vector 1 ∈ W the class 1 mod J ∈ V. Since all Bi′ s commute, then in−1 (B0i0 , . . . , Bn−1 , I) ∈ V(n, c). Moreover, the set n

i

n−1 B0i0 · B1i1 · · · Bn−1 I(1) ∈ V | ik = 0, . . . , c − 1

o

spans V as complex vector space. Therefore, by Lemma 3.4, the ADHM datum X is stable. To complete the proof, we only have to show that Ψ′ ◦ Ψ = 1M(n,c) and Ψ ◦ Ψ′ = 1Hilb[c] , i.e., the maps Ψ and Ψ′ are inverse to each other. Cn

Indeed, to each class [X] ∈ M(n, c), one associates the ideal Ψ([X]) = ker ΦX in Hilb[c] (Cn ). Moreover, one associates to the later ideal, ker ΦX , the ADHM datum ˜ Then one has [X] = [X] ˜ if and only if there exists an class Ψ′ (ker ΦX ) = [X]. ˜ element g ∈ GL(V ) such that X = g · X.

Let pr : C[Z0 , . . . , Zn−1 ] → C[Z0 , . . . , Zn−1 ]/ ker ΦX be the natural projection. From Lemma 3.2, it is clear that the diagram C[Z0 , . . . , Zn−1 ]

id

/ C[Z0 , . . . , Zn−1 ]

pr

ΦX

C[Z0 , . . . , Zn−1 ]/ ker ΦX

g

/V

commutes. Hence pr = g −1 ◦ ΦX , since g is an isomorphism. On the other ˜i ◦ pr : C[Z0 , . . . , Zn−1 ] → hand, from the Zi multiplication one has pr ◦ Zi = B [Z0 , . . . , Zn−1 ]/ ker ΦX and ΦX ◦ Zi = Bi ◦ ΦX : C[Z0 , . . . , Zn−1 ] → V. That is, one has the following diagram


9

pr

C[Z0 , . . . , Zn−1 ] id❥❥❥❥❥ ❥ u❥❥❥❥ ΦX C[Z0 , . . . , Zn−1 ]

/ Cc

Zi

/ C[Z0 , . . . , Zn−1 ]/ ker ΦX ❧ g ❧❧❧❧ ❧❧ ❧ ❧ ❧ ❧ u ❧ ˜ Bi

Bi

/ C[Z0 , . . . , Zn−1 ]/ ker ΦX ❧ g ❧❧❧❧ ❧ ❧ ❧ ❧❧❧ / C c u❧

C[Z0 , . . . , Zn−1 ] id❥❥❥❥❥ ❥ ❥ ❥ ❥ u❥ ΦX C[Z0 , . . . , Zn−1 ] Zi

pr

˜i ◦ g = Bi , for all i = {0, . . . , n − 1}. in which all faces commute. Then, one has g ◦ B ˜ Moreover, g ◦ I(1) = g(1 mod J) = ΦX (1) = I(1), i. e., g ◦ I˜ = I. Therefore, ˜ ∈ M(n, c), in other words, we have just shown that Ψ′ ◦ Ψ = 1M(n,c) . [X] = [X] [c]

To prove that Ψ◦Ψ′ = 1Hilb[c] , we only need to show that for a given J ∈ HilbCn , Cn

one has J = ker ΦX , where X is a ADHM datum in V(n, c)st such that Ψ′ (J) = [X] ∈ M(n, c). For a polynomial X αn−1 p(Z0 , . . . , Zn−1 ) = ∈ C[Z0 , . . . , Zn−1 ] aα Z0α0 · · · Zn−1 α

we have ΦX (p(Z0 , . . . , Zn−1 )) =

X

α

n−1 aα B0α0 . . . Bn−1 I(1)

α

where I(1) is the class 1( mod J) =: [1]. Moreover, since Bi ◦ ΦX = ΦX ◦ Zi then " # X X αn−1 αn−1 α0 α0 aα B0 . . . Bn−1 I(1) = aα Z0 . . . Zn−1 = [p(Z0 , . . . , Zn−1 )] . α

α

Thus, if the polynomial p(Z0 , . . . , Zn−1 ) belongs to the ideal J, then X αn−1 aα B0α0 . . . Bn−1 I(1) = 0, α

and therefore p(Z0 , . . . , Zn−1 ) ∈ ker ΦX . On the other hand, suppose that p(Z0 , . . . , Zn−1 ) ∈ ker ΦX . Then ΦX (p(Z0 , . . . , Zn−1 )) =

X

α

n−1 aα B0α0 . . . Bn−1 I(1) = 0.

α

Again, one has X α

aα B0α0

αn−1 I(1) . . . Bn−1

=

"

X α

aα Z0α0

αn−1 . . . Zn−1

#

= [p(Z0 , . . . , Zn−1 )] ,

hence [p(Z0 , . . . , Zn−1 )] = 0 ∈ C[Z0 , . . . , Zn−1 ]/J, that is, p(Z0 , . . . , Zn−1 ) ∈ J. Thus J = ker ΦX . This finishes our proof.

10


4. Extended monads and perfect extended monads In this section we shall generalize the concept of monads, introduced by Horrocks (the reader may consult [18] for definitions and properties), in order to describe ideal sheaves for zero-dimensional subschemes of Cn and Pn , n ≥ 2. Let X be a smooth projective algebraic variety of dimension n over the field of complex numbers C, and let OX (1) be a polarization on it. 4.1. l−extended monads. The objects we now wish to introduce are defined as follows. Definition 4.1. An l−extended monad over X is a complex C• :

(2)

α−l−1

α−l

α−1

α−2

α

0 C1 C −l−1 −→ C −l −→ · · · −→ C −1 −→ C 0 −→

of locally free sheaves over X which is exact at all but the 0−th position, i.e. Hi (C • ) = 0 for i 6= 0. The coherent sheaf E := H0 (C • ) = ker α0 /Im α−1 will be called a the cohomology of C • . Note that a monad on X, in the usual sense, is just a 0-extended monad. Moreover, one can associate to any l−extended monad C • a display of exact sequences as the following 0

0

C −l−1

C−l−1

(3)

α−l−1

α−l−1

C −l

C −l

α−l

.. .

α−l .. .

C −1

C −1

0

/K

/ C0

0

/F

/Q

0

0

α−2

α−2

α−1 α0

/ C1 / 0 / C1 / 0

where K := ker α0 and Q := cokerα−1 A morphism φ : C1• → C2• of two l−extended monads C1• and C2• is an (l + 3)−tuple of morphisms such that the following diagram commutes: (4)

C1• :

α1−l−1

C1−l−1

φ−l−1

φ

C2• :

/ C −l 1

α2−l−1 / C1−l−1

···

α1−2

/ C −1 1

φ−l

φ−1

C1−l

α1−1

···

α2−2

/ C −1 1

α2−1

/ C10

α10

φ0

/ C10

/ C11 φ1

α20

/ C21


11

With these definitions, the category of l−extended monads form a full subcategory of the category Kom♭ (X) of bounded complexes of coherent sheaves on X. l−extended monads have already appeared in the literature. The most important example of a locally-free sheaf that can be obtained as the cohomology of a 2-extended monad on P4 is the dual of the Horrocks–Mumford bundle; indeed, Fløystad shows in [4, Introduction: example b.] that the Horrocks–Mumford bundle is given by the cohomology at degree zero, where the grading is given by the twist, of a complex of the form OP54 (−1) → OP154 → OP104 (1) → OP24 (2). Dualizing such complex we get a 2-extended monad on P4 whose cohomology is the dual of the Horrocks–Mumford bundle. Moreover, object very closely related to 2-extended monads on P3 have also appeared in the mathematical physics literature, see [3, Section 4]. An l−extended monad can be broken into the following two complexes: first, (5)

N• :

0

/ C −l−1α−l−1 / C −l

α−3

···

/ C −2

α−2

/ C −1 J−1 / G

/0

which is exact, and a locally free resolution of the sheaf G = cokerα−2 , and (6)

M• :

G

I−1

/ C0

α0

/ C1

where I−1 ◦ J−1 = α−1 . M • is a monad-like complex in which the coherent sheaf G might not be locally free; indeed, G is not locally free for the extended monads describing ideal sheaves of 0-dimensional subschemes, the situation most relevant to the present paper. For a given l−extended monad, we refer to the complexes M • and N • as the associated resolution and the associated monad, respectively. Therefore, the morphism φ : C1• → C2• can be thought of as a pair of morphisms (φN : N1• → N2• , φM : M1• → M2• ) ∈ Hom(N1• , N2• ) × Hom(M1• , M1• ). −1 1 2 Remark that as long as we have φ0 (Im α−1 1 ) ⊆ Im α2 and φ0 (ker α0 ) ⊆ ker α0 then φ is determined by only φ0 ; indeed, the conditions −1 φ0 (Im α−1 1 ) ⊆ Im α2

and

φ0 (Im I1−1 ) ⊆ Im I2−1

are equivalent (here we considered the morphism of the associated monads). Hence φ0 determines the morphism φM , and consequently it also determines the whole morphism φ : C1• → C2• . This is because N1• and N2• are locally free resolutions and hence projective resolutions, so that giving a morphism φG : G → G determines all the morphisms φ−i : C1−i → C2−i for i ≤ −1. Since taking cohomology is a functorial operation, a morphism φ : C1• → C2• of two l−extended monads C1• and C2• , induces a morphism between their respective cohomologies H(φ) : H0 (C1• ) → H0 (C2• ). Of course, isomorphic complexes induce isomorphic cohomologies. It follows that there is natural notion of equivalence for l−extended monads with the same terms C i provided by the action of the automorphism group Aut(C • ) = Aut(C −l−1 ) × Aut(C −l ) × · · · × Aut(C 0 ) × Aut(C 1 ). Our goal now is to study families of ideal sheaves of zero-cycles in Pn . It turns out that such ideal sheaves are given by cohomologies of a special kind of l−extended monads. However, before proving this claim, which will be done only in Section

12


5 below, we tackle a more general question, namely under which conditions a homomorphism H0 (C1• ) → H0 (C2• ) lifts to a homomorphism C1• → C2• between the corresponding complexes. In particular to determine the automorphisms of such objects. Our next result provides a sufficient condition, by showing when the cohomology functor is full and faithful. Proposition 4.2. Let α1−l−1

C1−l−1

C1• :

C2−l−1

C2• :

/ C −l 1

···

α2−l−1

/ C 2−n 2

α1−2

α1−1

/ C −1 1

···

α2−2

/ C10

/ C −1 2

α2−1

α10

/ C11

/ C20

α20

and / C21

be two l-extended monads, and let us denote by F1 and F2 their respective cohomologies. Then H : Hom(C1• , C2• ) → Hom(F1 , F2 ) is surjective if Ext1 (C11 , C10 ) = 0, Extk (C10 , C2−k ) = 0 for k ≥ 1, Extk (C11 , C2−k+1 ) = 0 for k ≥ 2. Moreover if Hom(C11 , C20 ) = 0, Extk (C10 , C2−k+1 ) = 0 for k ≥ 1, then H is an isomorphism.

Extk (C10 , C2−k−1 ) = 0 for al k ≥ 0,

Proof. Let G1 = Im α1−1 and G2 = cokerα2−1 . The associated resolution N2• can be broken into sequences 0 → G2−i → C2−i → G2−i+1 → 0,

1≤i≤l+1

where we put G2−l−1 = C2−l and G20 = G2 . Then, by applying either Hom(C10 , •) or Hom(C11 , •) on the above sequences and incorporating the conditions given in the proposition, it follows that H is surjective if Ext1 (C11 , C10 ) = Ext1 (C10 , G2 ) = Ext2 (C11 , G2 ) = 0, and it is an isomorphism if Hom(C11 , C20 ) = Hom(C10 , G2 ) = Ext1 (C10 , G2 ) = 0. To finish the proof, it suffice to apply [18, Lemma 4.1.3] to the associated monad M2• of the l−extended monad C2• . Corollary 4.3. Let C• :

α−l−1

C −l−1

/ C −l

···

α−2

/ C −1

α−1

/ C0

α0

/ C1

and α′

α′

α′

α′

−l−1 −2 / C −l / C −1 −1 / C 0 0 / C 1 C ′• : C −l−1 ··· be l-extended monads, and let F and F ′ be their cohomologies, respectively. Suppose that Ext1 (C 1 , C 0 ) = 0,

Extk (C 0 , C −k ) = 0 for k ≥ 1,

Extk (C 1 , C −k+1 ) = 0 for k ≥ 2.


13

Then F and F ′ are isomorphic if and only if C • and C ′• are isomorphic (as lextended monads). 4.2. Perfect extended monads. We now introduce the class of l−extended monads which is relevant to the description of the Hilbert scheme of points. ⊕a

Definition 4.4. An l−extended monad C • is called pure if C −i = L−i −i , for all −1 ≤ i ≤ l − 1, where L−i , are invertible sheaves, and it is called linear if all maps α−i are given by matrices of linear forms. Before our next definition, recall that OX (1) is the chosen polarization on the n−dimensional projective algebraic variety X. Definition 4.5. A perfect extended monad on a n-dimensional projective variety X is a linear (n − 2)−extended monad P • on X of the following form OX (1 − n)⊕a1−n ···

α1−n

/ OX (2 − n)⊕a2−n

/ OX (−1)⊕a−1 α−1 / O⊕a0 X

α−2

/ α0

/ OX (1)⊕a1 .

We recall to the reader that a projective scheme X is arithmetically CohenMacaulay, or simply ACM, if its homogeneous coordinate ring is Cohen-Macaulay ring. Moreover let us denote by Per the full subcategory of Kom♭ (X) consisting of perfect extended monads. Corollary 4.6. If X is an n−dimensional ACM variety, then the cohomology functor H : Pre(X) → Coh(X) is full and faithfull. Proof. This follows easily from Proposition 4.2: since X is ACM, we have that Hom(C11 , C20 ) = H0 (X, OX (−1)) = 0

and

Exti (OX (a), OX (b)) = Hi (X, OX (b − a)) = 0, for 1 ≤ i ≤ n − 1. It follows from the Corollary above that automorphism group of a perfect extended monad on an ACM variety is just GLa1−n (C) × GLa−n (C) × · · · × GLa1 (C). We finish this section by describing the cohomology of sheaves which arise as cohomologies of perfect extended monads on Pn , n ≥ 2. Proposition 4.7. If F is the cohomology of a perfect extended monad on Pn (n ≥ 2) then: (i) H0 (F (k)) = 0 for k < 0; (ii) Hn (F (k)) = 0 for k > −n − 1; (iii) Hi (F (k)) = 0 ∀k, 2 ≤ i ≤ n − 1, when n ≥ 3.

14


Proof. We twist the middle column of the display (3) by OPn (k), then break it into short exact sequences (7)

0

/ OPn (k + 1 − n)⊕a1−n

/ OPn (k + 2 − n)⊕a2−n

/ Q2−n (k)

/0

0

/ Q2−n (k)

/ OPn (k + 3 − n)⊕a2−n

/ Q3−n (k)

/0

/ Q−p (k)

/0

.. . 0

/ Q−p−1 (k)

/ OPn (k − p)⊕a−p .. .

0

/ Q−2 (k)

/ OPn (k − 1)⊕a−1

/ Q−1 (k)

/0

0

/ Q−1 (k)

/ OPn (k)⊕a0

/ Q0 (k)

/0

where Q0 := Q = cokerα−1 . Step. 1: From the long sequences in cohomology of the first row above, we have Hi (OPn (k + 2 − n))⊕a2−n → Hi (Q2−n (k)) → Hi+1 (OPn (k + 1 − n))⊕a1−n → · · · Then, from the vanishing properties of line bundles on Pn , it follows that (i) H0 (Q2−n (k)) = 0 for k < n − 2; (ii) Hn (Q2−n (k)) = 0 for k > −1; (iii) Hi (Q2−n (k)) = 0 ∀k, 1 ≤ i ≤ n − 1. Step. 2: Using induction on the remaining rows in (7) it follows that, for p > 2, (i) H0 (Qp−n (k)) = 0 for k < n − p; (ii) Hn (Qp−n (k)) = 0 for k > −p − 1; (iii) Hi (Qp−n (k)) = 0 ∀k, 1 ≤ i ≤ n − 1. Step. 3: From the long exact sequence in cohomology of the lower row in (3) twisted by OPn (k) one has Hi−1 (OPn (k + 1))⊕a1 → Hi (F (k)) → Hi (Q(k)) → · · · Using the vanishing obtained in Step. 2 for Q0 = Q, the claims of items (i), (ii), (iii) and (iv) follow. The last item is obtained by dualizing the lower row of (3). n Now let us denote by Ω−p Pn the bundle of holomorphic (−p)−forms on P , where p ≤ 0 in our convention.

Proposition 4.8. If a coherent sheaf E on Pn (n ≥ 2) satisfies: (i) H0 (E(−1)) = Hn (Pn , E(−n)) = 0; (ii) Hq (Pn , E(k)) = 0 ∀k, 2 ≤ q ≤ n − 1 when n ≥ 3; (iii) H1 (Pn , E ⊗ Ω−p Pn (−p − 1)) 6= 0 for −n ≤ p ≤ 0; then E is the cohomology of a perfect extended monad.


15

Proof. Applying Beilinson’s theorem [18, Theorem 3.1.4] to the sheaf E(−1), one gets a spectral sequence with E1 −term given by E1p,q = Hq (E ⊗ Ω−p Pn (−p − 1)) ⊗ OPn (p) which converges to the graded sheaf associated to a filtration of E(−1) itself. Twist the Euler sequence for the sheaves of differential forms n+1 0 → Ωp (p) → OPNn → Ωp−1 (p) → 0 , N = p by E(k−p) and use hypotheses (i) and (ii) above to conclude, after long but straightforward calculations with the associated long exact sequences of cohomology, that E1p,q = 0 for q 6= 1. It follows immediately that the Beilinson spectral sequence degenerates already at the E2 -term, i.e. E2 = E∞ . Beilinson’s theorem then implies that the complex E1p,1 given by Vn ⊗ OPn (−n) → · · · → V1 ⊗ OPn (−1) → V0 ⊗ OPn ,

(8)

with Vp := H (E ⊗ Ω−p Pn (−p − 1)), −n ≤ p ≤ 0, is exact everywhere except at position p = −1, and its cohomology at this position is precisely E(−1). The third hypothesis implies that none of the vector spaces Vp vanishes. So twisting the complex (8) by OPn (1), we obtain a perfect extended monad whose cohomology is exactly E, as desired. 1

5. Ideal sheaves of zero-dimensional subschemes of Pn We now consider sheaves E of rank r on Pn fitting in the following short exact sequence (9)

0 → E → OP⊕r n → Q → 0,

where Q is a pure torsion sheaf of length c supported on a 0-dimensional subscheme Z ⊂ Pn . Note that the Chern character of E is given by ch(E) = r − cH n , and that E is necessarily torsion free. Such sheaves can also be regaded as points in the Quot scheme QuotP =c (OP⊕r n ). In the case r = 1, it is clear that E is the sheaf of ideals in OPn associated to the zero-dimensional subscheme Z, i.e. Q = OZ ; in this case, we will then denote E by IZ . Proposition 5.1. Every sheaf E on Pn given by sequence (9) is the cohomology of a perfect extended monad P • with terms of the form P −i := Vi ⊗ OPn (i), i = 1 − n, . . . , 0, 1, where 1−i ∼ 0 (10) Vi := H1 (E ⊗ Ω1−i n (−i)) = H (Q ⊗ Ω n (−i)). P

P

Furthermore, we the following isomorphisms: (11) V1 ∼ = H0 (Q) (12)

Vi ∼ =

(

V1⊕n ⊕ Cr ⊕( n ) V1 1−i

for i = 0 for i < 0

16


In particular, we conclude that dim Vi =

 

c nc + r  n 1−i c

i=1 i=0 i