Quivers and equations a la Plücker for the Hilbert scheme

Quivers and equations a la Plücker for the Hilbert scheme

Laurent Evain and Margherita Roggero

Abstract:

Several moduli spaces parametrising linear subspaces of the projective space admit a natural projective embedding in which they are cut out by linear and quadratic equations (Grassmannians, flag varieties, and Schubert varieties). The aim of this paper is to prove that a similar statement holds when one replaces linear subspaces with algebraic subschemes of the projective space. We exhibit equations of degree 1 and 2 that define schematically the Hilbert schemes for all (possibly non-constant) Hilbert polynomials . The equations are reminiscent of the Plücker relations on the Grassmannians: they are built formally with wedge products and permutations on indexes on the Plücker coordinates. Our method relies on a new description of the Hilbert scheme as a quotient of a scheme of quiver representations.

1. Introduction

The Plücker coordinates on a Grassmannian satisfy the well known Plücker relations. Similarly, the flag varieties are defined by quadratic equations and Schubert varieties are defined by quadratic and linear equations [23, 8]. The Grassmannians, flag varieties and Schubert varieties parametrize linear subspaces in a projective space. The goal of this paper is to prove that analog results hold in a non-linear context. We consider the Hilbert schemes parametrising the algebraic subschemes of a projective space and we prove that they are defined by simple explicit linear and quadratic equations in their natural embedding.

The Hilbert schemes carry in general a natural non-reduced structure inherited from their functorial construction. Our equations take into account the non-reduced structure and define the Hilbert schemes schematically.

More specifically, let be the Hilbert scheme parametrising closed subschemes of with Hilbert polynomial over a field . There is a Grassmannian embedding , where is any integer larger or equal to the Castelnuovo-Mumford-Gotzmann number of and . Composing with the Plücker embedding , , we consider the problem of finding equations for the Hilbert scheme in .

The question of finding equations for the Hilbert scheme as a subscheme of a Grassmannian has been addressed many times after its introduction by Grothendieck. The equations that arise depend much on the way the Hilbert scheme is described. The initial construction of the Hilbert scheme involved flattening stratifications [15, Lemme 3.4]. Techniques were developed to compute local equations for the flat stratum corresponding to the Hilbert scheme [9][12, Proposition 0.5]. The work by Gotzmann [11] leads to a description of the Hilbert scheme as a determinantal locus. This determinantal approach was also used by Bayer in his PhD thesis [3] to build up a set of equations defining set theoretically the Hilbert scheme. It was proved by Iarrobino and Kleiman [19, Appendix +C] exploiting an argument of Grothendieck that the Bayer equations hold scheme theoretically. Haiman and Sturmfels obtained the Bayer equations schematically as a special case of their own construction of the multigraded Hilbert scheme [17]. In [4] and [20], Brachat, Lella, Mourrain and Roggero define the Hilbert scheme using a functor which involves the action of to use the symmetry of the Hilbert scheme. See also [21] for techniques using Border bases.

The various approaches lead to equations of different degrees: for instance degree , only depending on the “ambient” space , for those by Bayer, Iarrobino-Kleiman and Haiman-Sturmfels, degree , only depending on the Hilbert polynomial, for those by Brachat-Lella-Mourrain-Roggero.

We will see that it is possible to find equations of degree 1 and 2 that cut out the Hilbert scheme when is non-constant, and equations of degree when is constant. These are obviously the smallest possible degrees since in general the Hilbert scheme is not a linear space, not even a linear section of a Grassmannian [4, Section 7.2].

It was remarked by Haiman and Sturmfels [17] that their quite theoretical construction of the Hilbert scheme provides access to equations hardly accessible by direct computation. In cryptography, systems built with rich structures are possibly fragile because attackers may extract information from the structure. The above list of examples suggest that a similar principle could hold in our context : a new description of the Hilbert scheme could reveal a structure providing access to some new equations.

Starting from these remarks, our approach is to produce a new description for the Hilbert scheme and to extract equations of small degree from the construction.

We consider the description by Nakajima of , when is a constant polynomial. It is related to the framed moduli space of torsion free sheaves on , monads and adhm-structures, quivers or commuting matrices [22]. We seek a description in the same vein for , i.e. we want to replace the constant by any polynomial and the affine plane by a projective space of any dimension.

An extension of Nakajima’s description has been realized by Bartocci, Bruzzo, Lanza and Rava in [2]. They replaced the affine plane with the total space of . They use a description of the moduli space parametrising isomorphism classes of framed sheaves on the Hirzebruch surface . The computations of the paper show that it is not possible to extend the initial description by Nakajima directly. In the sheaf context, the trivialization at infinity of the sheaf is responsible for the loss of projectivity. Replacing the surface by a higher dimensional variety or considering a non-constant Hilbert polynomial weakens the link between sheaves and Hilbert schemes.

We may reformulate the above difficulties in matrix terms. Recall that a zero-dimensional subscheme is represented by a pair of commuting matrices corresponding to the multiplication by the variables on the vector space , together with a cyclic vector for the pair . The matrices are determined up to the choice of the base of , and the cyclic vector is the algebraic counterpart of the constant function generating as a -module. In a nutshell, the Hilbert scheme is constructed as a -quotient of an open set of a commuting variety parametrising pairs of commuting matrices.

Considering now any subscheme with Hilbert polynomial , we try to characterize using matrices corresponding to multiplication by the variables, up to the choice of the base. The multiplication by the variable yields a morphism , where is chosen fixed and larger than or equal to the Castelnuovo-Mumford regularity of . However, the source space and the target space are different and the commutativity does not make sense. When is non-constant, the underlying matrices are not square matrices and their size are incompatible. When is constant, the matrix sizes are compatible but we miss a trivialization at infinity to identify with . Finally, in the affine case, the constant function generates as a -module. In the projective case, there is no privileged element in and no natural cyclic vector notion.

The above analysis shows that for a description of based on the multiplicative action of the variables, we require a framework where we can formulate substitute conditions for the commutativity and cyclic conditions. In the first part of the paper, we introduce a quiver and we formulate these substitutes as technical conditions on the representations of the quivers that we consider. We proceed as follows.

We choose any integer larger than or equal to the Gotzmann number of and we consider the quiver with 4 vertices, arrows, dimension vector and corresponding vector spaces , where .

Then we consider the representations of the quiver such that:

  • The map is the multiplication by the variable .

  • The map is surjective

  • The images of the satisfy the condition .

  • for every .

There is a natural functor associated to the above representations, which is represented by a scheme . There is an action of on corresponding to the base changes on the last two vertices of the quiver. Our description of the Hilbert scheme is summarized in the following theorem.

Theorem 1.1.

is a principal bundle over the Hilbert scheme .

The theorem provides a new universal property for the Hilbert scheme: it is possible to describe locally a family of subschemes of using families of matrices from the quiver description, up to action of the group. Describing schemes in terms of linear algebra up to action may be more convenient than the usual description in terms of polynomial ideals (see [5, Prop. 3.14] for an explicit example).

Recall that Grassmannians are quotients of Stiefel varieties, and that Plücker coordinates are computable from Stiefel coordinates [10]. In our context, the “Stiefel” coordinates are on , they are the entries of the matrices . The following proposition describes similarly the Plücker coordinates of the Hilbert schemes in terms of the “Stiefel coordinates of ”.

Proposition 1.2.

The Plücker coordinates in are the maximal minors of . The Plücker coordinates in are the maximal minors of .

The notations to formulate our equations are as follows. If is a variable, if are monomials, is a product of monomials in and it is a Plücker coordinate on the Grassmannian . If is the generic linear form with indeterminate coefficients , the multilinear expansion of is a linear combination of Plücker coordinates. This expansion is a polynomial in the variables and we denote by the coefficients of this polynomial. Similarly, we denote by symbols the coefficients of the expansion of . Both and are linear combinations of Plücker coordinates on .

Theorem 1.3.

Suppose that is a non-constant Hilbert polynomial, its Gotzmann number, , and consider the composed embedding . Let be the ideal generated by:

  • the quadratic Plücker relations of the Grassmannian,

  • the linear forms

  • the quadrics

Then is the subscheme defined by the ideal .

When is constant, we have the same result as above, except that the set of linear forms is empty. Thus is generated by the quadrics of the first and third items in the list.

There is always an ambiguity for the signs of the Plücker coordinates. Our convention in these equations is to consider Plücker coordinates of the quotient.

Overview of the proof of Theorem 1.3

In the first part of the proof, the equations are obtained as a direct algebraic consequence of our quiver description. Recall that the Plücker coordinates in degree appear as determinants of by Proposition 1.2. When is non-constant, the composition is not surjective for obvious dimensional reasons and we get the vanishing of the corresponding determinant/Plücker coordinate. After a few algebraic manipulations to get the maximum from this idea, we get the linear equations .

In the second part of the proof, we investigate the geometrical meaning of these algebraic vanishings. We characterize the locus defined by the equations in terms of locally free sheaves (Proposition 6.7).

More specifically, let be the locus in the Grassmannian cut out by all the linear equations . A closed point parametrises a vector space and we consider for any linear form . For any and a general , . If , we have the equality . The set of linear forms such that the inequality holds depend on . To work functorially with families, a linear form suitable for all simultaneously is necessary. To bypass this difficulty, we follow Grothendieck and we use non-closed points: the generic linear form with indeterminate coefficients may be used uniformly for all . Technically, we work over the residual field and we show that is a locally free sheaf of rank on some nice open set (6.7).

The next step is to compare this geometrical interpretation in terms of locally free sheaves to the Gotzmann-Iarrobino-Kleiman description of the Hilbert scheme. The analysis of the difference leads to the missing quadratic equations , as follows.

Let . By the above, iff for general while Gotzmann’s description says that iff . Heuristically, if is generic and is general, we have the following sequence of inclusion:

Indeed, if satisfies for the generic then for every linear form specialization of . The left inclusion follows and the right inclusion is obvious. When , and have the same codimension , we conclude that and by Gotzmann’s description, .

This heuristic is not correct since it is careless about the residual fields. In general, is a -point (i.e. the computation yields a formula depending on the coefficients of the generic form ) whereas is a -point. However, if is a -point, the above reasoning makes sense and this yields an equivalence: if , then if and only if is a -point. (Proposition 7.1).

It remains to prove that this condition on the base field of corresponds to the quadratic equations (Proposition 7.7). To settle this, we compute the (superabundant) Plücker coordinates of which are elements in (Proposition 7.5). The formula obtained and the simple cross product remark 7.6 show that is a -point exactly when the quadratic equations hold.

A workshop “Components of Hilbert Schemes” was organized by the American Institute of Mathematics from July 19 to July 23, 2010. This is the place where the authors met for the first time. We thank the institute and the organizers.
We thank Steve Kleiman and Michel Brion for their useful comments.

2. Embeddings of the Hilbert scheme

Until section 8, we work over a field algebraically closed of arbitrary characteristic. We will prove in section 8 that our equations are valid on any field.

In this section, we recall some of the classical material used to embed Hilbert schemes into Grassmannians.

Notation 2.1.

Let , and for any -algebra . We denote by and the free submodules of homogeneous polynomials of degree . We denote by the same letter and the multiplication by the variable .

Recall [13, p.80] that if is the Hilbert polynomial of a subscheme , then

where the number of binomials is called the Gotzmann number of . It depends on , but not on . For every and every , the Hilbert function and the Hilbert polynomial satisfy [19, Corollary C.15]. The Gotzmann number coincides with the Castelnuovo-Mumford regularity of , i.e. the smallest integer such that every with Hilbert polynomial is -regular [19, Proposition C.24].

From now on, will denote a Hilbert polynomial for subschemes of , its Gotzmann number and any number .

We recall the following result which realizes as a closed subscheme of the product of Grassmannians [11, Bemerkung 3.2],[19, Proposition C.28, Theorem C.29], [17].

Theorem 2.2.

The Hilbert scheme represents a functor from -algebras to sets where with , and locally free -submodules of rank and respectively, and for each variable , . In particular, is a closed subscheme of the product of Grassmannians .

Moreover, the first (resp. second) projection gives an embedding (resp. ).

Let us fix any positive integer . The binomial expansion of a positive integer in base , also called the -th Macaulay representation of , is the unique expression

with . Let

and

Theorem 2.3.

(Macaulay)[13, p.79]. Let with codimension , the image of the multiplication , and the codimension of . Then .

Theorem 2.4.

(Green)[13, p.77]. Let with codimension , the restriction of to a general hyperplane , and the codimension of . Then .

Theorem 2.5.

Let be a subscheme with Hilbert polynomial and consider a degree . Then , i.e. .

Proof.

In degree , the codimension is computed by the Hilbert polynomial, whose value is with according to the Gotzmann regularity Theorem [13, p.80]. The relation between and follows immediately. ∎

3. Description of the Hilbert scheme

In this section, we give the description of the Hilbert scheme in terms of representation of quivers.

Notation 3.1.

If , for , are morphisms of -modules and is a morphism of -algebras we will use the following notations

  • is the morphism of modules with ,

  • is the list ,

  • is the morphism given by ,

  • is the morphism given by .

We recall the quiver from the introduction.

To build the variety above the Hilbert scheme, a subset of representations of the quiver is considered. The following definition introduces these representations in a functorial way.

Definition 3.2.

Let be a -algebra. Let where:

  • and is the multiplication by the variable ,

  • and is a morphism of -modules,

  • is surjective,

  • is a surjective morphism of -modules,

  • for every pair , .

Remark 3.3.

The set and the map depend on , but for brevity is not included in our notation. Similarly, we will use the notation as a shortcut for since there is only one possible choice for .

Since the tensorisation preserves the surjectivity, for any map of -algebras , we have a morphism which sends to . This makes a functor from the category of -algebras to the category of sets.

Proposition 3.4.

There exists a scheme such that:

  • the -points of are representations of the quiver .

Proof.

The non-trivial fact is the first item. It follows immediately that the -points are representations of .

Let be the extension of to the category of -schemes, i.e. where:

  • and is the multiplication by the variable ,

  • and is a morphism of -modules,

  • is surjective

  • is a surjective morphism of -modules,

  • for every pair , .

It suffices to prove that is representable to obtain the first item of the proposition.

Consider the functor defined as follows. If is a -scheme, an element of is a couple where:

  • is a morphism of -modules,

  • is a (possibly not surjective) morphism of -modules.

For any map of -schemes , we have a morphism which sends to .

For any finite dimensional -vector space , let us denote by the functor defined by and, for any map of -schemes , the map sends to by pullback. It is well known that is represented by , the scheme associated to . In particular, is represented by with .

We recall the notion of relative representability from [14]. Let be functors from the category of -schemes to sets. Suppose that is a subfunctor of . The inclusion is relatively representable if, for every -scheme with functor , and every morphism of functors , the cartesian product is representable. Grothendieck, [14, Lemme 3.6] proves that if is representable and if is relatively representable, then is representable.

In our case, is representable and its subfunctor is defined by the surjectivity of and , and by the equality . Thus it suffices to prove that a subfunctor defined by the surjectivity of a morphism of locally free sheaves is relatively representable, and that a subfunctor defined by the equality of morphisms of locally free sheaves is relatively representable.

The locus in where two matrices of size with coefficients in coincide is closed. Indeed, if is a morphism, then the pullback matrices satisfy if and only if the morphism factorizes through the closed subscheme where the ideal is generated by the elements . It follows that if are locally free sheaves on , and if are two morphisms of sheaves, there exists a closed subscheme , such that for all , iff factorizes through . Let be a functor such that , i.e. is a tuple, and two components of this tuple correspond to a morphism of locally free sheaves naturally associated to . Let be the subfunctor of defined by the condition . By Yoneda, a morphism is defined by an element . By the above, can be identified with the set of morphisms . Thus and is a relatively representable functor. It follows that the condition defines a relatively representable (closed) subfunctor of .

The fact that the surjectivity condition on a morphism of sheaves defines an open subfunctor is a classical argument used in the construction of the Grassmannians [16, Lemme 9.7.4.6].

Thus is representable as it is a locally closed subfunctor of the representable functor . ∎

We denote by the group of invertible matrices with coefficients in and we use the abbreviation .

Proposition 3.5.

There is an action of the group on .

Proof.

At the functorial level, if is an -point of and are two matrices with , the action is defined by with and . ∎

Our goal is to prove that the Hilbert scheme is a geometric quotient of by the above natural action. We start with the construction of a morphism.

Proposition 3.6.

There exists a morphism . More specifically, if is an -point of , then is the -point of the Hilbert functor defined by the ideal generated by .

Proof.

We specify the notation, all appearing in the diagram 3.1. Since the argument will involve several degrees, we let instead of for the multiplications . Let ( copies ).

We consider the maps , and obtained from , following the conventions introduced in Notation 3.1. For readability, we let . Moreover, we denote by the restriction of to , and we let .

Claim: We can define a morphism such that the following diagram is commutative with exact rows.

(3.1)

We observe that

  • , , and are surjective by hypotheses and/or by construction,

  • by construction the first row is exact and the square on the left commutes.

We use all these properties in order to define the dash arrow so that also the last line is exact and all the diagram commute.

We define by diagram chasing in the following way: by the surjectivity of every element of can be written (not uniquely) as where ; then we set .

To verify that is well defined, we prove that when we have . This is obvious if , since implies . Then, we prove the assertion for assuming it holds for elements of the form .

For every we set with and not appearing in . The equality implies and . Then we have

The last summand is equal to , hence it vanishes by the inductive assumption. Moreover, by the commutativity conditions in the definition of , we have . Therefore

The commutativity of the right square holds by the construction of and the surjectivity of is a direct consequence of that of , and and of the commutativity of the right square.

To complete the construction of our diagram, we now prove that is equal to . By the commutativity of the two squares and the surjectivity of