# Hilbert schemes, polygraphs and the Macdonald positivity conjecture

By Mark Haiman

## Abstract

We study the isospectral Hilbert scheme , defined as the reduced fiber product of with the Hilbert scheme of points in the plane , over the symmetric power . By a theorem of Fogarty, is smooth. We prove that is normal, Cohen-Macaulay and Gorenstein, and hence flat over . We derive two important consequences.

(1) We prove the strong form of the conjecture of Garsia and the author, giving a representation-theoretic interpretation of the Kostka-Macdonald coefficients . This establishes the Macdonald positivity conjecture, namely that .

(2) We show that the Hilbert scheme is isomorphic to the -Hilbert scheme of Nakamura, in such a way that is identified with the universal family over . From this point of view, describes the fiber of a character sheaf at a torus-fixed point of corresponding to .

The proofs rely on a study of certain subspace arrangements , called polygraphs, whose coordinate rings carry geometric information about . The key result is that is a free module over the polynomial ring in one set of coordinates on . This is proven by an intricate inductive argument based on elementary commutative algebra.

## 1. Introduction

The Hilbert scheme of points in the plane is an algebraic variety which parametrizes finite subschemes of length in . To each such subscheme corresponds an -element multiset, or unordered -tuple with possible repetitions, of points in , where the are the points of , repeated with appropriate multiplicities. There is a variety , finite over , whose fiber over the point of corresponding to consists of all ordered -tuples whose underlying multiset is . We call the isospectral Hilbert scheme.

By a theorem of Fogarty Reference14, the Hilbert scheme is irreducible and nonsingular. The geometry of is more complicated, but also very special. Our main geometric result, Theorem 3.1, is that is normal, Cohen-Macaulay and Gorenstein.

Earlier investigations by the author Reference24 unearthed indications of a far-reaching correspondence between the geometry and sheaf cohomology of and on the one hand, and the theory of Macdonald polynomials on the other. The Macdonald polynomials

are a basis of the algebra of symmetric functions in variables , with coefficients in the field of rational functions in two parameters and . They were introduced in 1988 by Macdonald Reference39 to unify the two well-known one-parameter bases of the algebra of symmetric functions, namely, the Hall-Littlewood polynomials and the Jack polynomials (for a thorough treatment see Reference40). It promptly became clear that the discovery of Macdonald polynomials was fundamental and sure to have many ramifications. Developments in the years since have borne this out, notably, Cherednik’s proof of the Macdonald constant-term identities Reference9 and other discoveries relating Macdonald polynomials to the representation theory of quantum groups Reference13 and affine Hecke algebras Reference32Reference33Reference41, the Calogero-Sutherland model in particle physics Reference35, and combinatorial conjectures on diagonal harmonics Reference3Reference16Reference22.

The link between Macdonald polynomials and Hilbert schemes comes from work by Garsia and the author on the Macdonald positivity conjecture. The Schur function expansions of Macdonald polynomials lead to transition coefficients , known as Kostka-Macdonald coefficients. As defined, they are rational functions of and , but conjecturally they are polynomials in and with nonnegative integer coefficients:

The positivity conjecture has remained open since Macdonald formulated it at the time of his original discovery. For it reduces to the positivity theorem for -Kostka coefficients, which has important algebraic, geometric and combinatorial interpretations Reference7Reference10Reference17Reference27Reference31Reference34Reference36Reference37Reference38Reference45. Only recently have several authors independently shown that the Kostka-Macdonald coefficients are polynomials, , but these results do not establish the positivity Reference18Reference19Reference32Reference33Reference44.

In Reference15, Garsia and the author conjectured an interpretation of the Kostka-Macdonald coefficients as graded character multiplicities for certain doubly graded -modules . The module is the space of polynomials in variables spanned by all derivatives of a certain simple determinant (see §2.2 for the precise definition). The conjectured interpretation implies the Macdonald positivity conjecture. It also implies, in consequence of known properties of the , that for each partition of , the dimension of is equal to . This seemingly elementary assertion has come to be known as the conjecture.

It develops that these conjectures are closely tied to the geometry of the isospectral Hilbert scheme. Specifically, in Reference24 we were able to show that the Cohen-Macaulay property of is equivalent to the conjecture. We further showed that the Cohen-Macaulay property of implies the stronger conjecture interpreting as a graded character multiplicity for . Thus the geometric results in the present article complete the proof of the Macdonald positivity conjecture.

Another consequence of our results, equivalent in fact to our main theorem, is that the Hilbert scheme is equal to the -Hilbert scheme of Ito and Nakamura Reference28, for the case , . The -Hilbert scheme is of interest in connection with the generalized McKay correspondence, which says that if is a complex vector space, is a finite subgroup of and is a so-called crepant resolution of singularities, then the sum of the Betti numbers of equals the number of conjugacy classes of . In many interesting cases Reference6Reference42, the -Hilbert scheme turns out to be a crepant resolution and an instance of the McKay correspondence. By our main theorem, this holds for , .

We wish to say a little at this point about how the discoveries presented here came about. It has long been known Reference27Reference45 that the -Kostka coefficients are graded character multiplicities for the cohomology rings of Springer fibers. Garsia and Procesi Reference17 found a new proof of this result, deriving it directly from an elementary description of the rings in question. In doing so, they hoped to reformulate the result for in a way that might generalize to the two-parameter case. Shortly after that, Garsia and the author began their collaboration and soon found the desired generalization, in the form of the conjecture. Based on Garsia and Procesi’s experience, we initially expected that the conjecture itself would be easy to prove and that the difficulties would lie in the identification of as the graded character multiplicity. To our surprise, however, the conjecture stubbornly resisted elementary attack.

In the spring of 1992, we discussed our efforts on the conjecture with Procesi, along with another related conjecture we had stumbled upon in the course of our work. The modules involved in the conjecture are quotients of the ring of coinvariants for the action of on the polynomial ring in variables. This ring is isomorphic to the space of diagonal harmonics. Computations suggested that its dimension should be and that its graded character should be related to certain well-known combinatorial enumerations (this conjecture is discussed briefly in §5.3 and at length in Reference16Reference22). Procesi suggested that the Hilbert scheme and what we now call the isospectral Hilbert scheme should be relevant to the determination of the dimension and character of . Specifically, he observed that there is a natural map from to the ring of global functions on the scheme-theoretic fiber in over the origin in . With luck, this map might be an isomorphism, and—as we are now able to confirm— might be flat over , so that its structure sheaf would push down to a vector bundle on . Then would coincide with the space of global sections of this vector bundle over the zero-fiber in , and it might be possible to compute its character using the Atiyah-Bott Lefschetz formula.

The connection between and the conjecture became clear when the author sought to carry out the computation Procesi had suggested, assuming the validity of some needed but unproven geometric hypotheses. More precisely, it became clear that the spaces in the conjecture should be the fibers of Procesi’s vector bundle at distinguished torus-fixed points in , a fact which we prove in §3.7. These considerations ultimately led to a conjectured formula for the character of in terms of Macdonald polynomials. This formula turned out to be correct up to the limit of practical computation (). Furthermore, Garsia and the author were able to show in Reference16 that the series of combinatorial conjectures in Reference22 would all follow from the conjectured master formula. Thus we had strong indications that Procesi’s proposed picture was indeed valid, and that a geometric study of should ultimately lead to a proof of the and Macdonald positivity conjectures, as is borne out here. By now the reader should expect the geometric study of also to yield a proof of the character formula for diagonal harmonics and the conjecture. This subject will be taken up in a separate article.

The remainder of the paper is organized as follows. In §2 we give the relevant definitions concerning Macdonald polynomials and state the positivity, and graded character conjectures. Hilbert scheme definitions and the statement and proof of the main theorem are in §3, along with the equivalence of the main theorem to the conjecture. In §3.9 we review the proof from Reference24 that the main theorem implies the conjecture of Garsia and the author on the character of the space , and hence implies the Macdonald positivity conjecture.

The proof of the main theorem uses a technical result, Theorem 4.1, that the coordinate ring of a certain type of subspace arrangement we call a polygraph is a free module over the polynomial ring generated by some of the coordinates. Section 4 contains the definition and study of polygraphs, culminating in the proof of Theorem 4.1. At the end, in §5, we discuss other implications of our results, including the connection with -Hilbert schemes, along with related conjectures and open problems.

## 2. The n factorial $n!$ and Macdonald positivity conjectures

### 2.1. Macdonald polynomials

We work with the transformed integral forms of the Macdonald polynomials, indexed by integer partitions , and homogeneous of degree . These are defined as in Reference24, eq. (2.18), to be

where denotes Macdonald’s integral form as in Reference40, VI, eq. (8.3), and is the partition statistic

(not to be confused with ).

The square brackets in Equation3 stand for plethystic substitution. We pause briefly to review the definition of this operation (see Reference24 for a fuller discussion). Let be the algebra of formal series over the coefficient field , in variables . For any , we denote by the result of replacing each indeterminate in by its -th power. This includes the indeterminates and as well as the variables . The algebra of symmetric functions is freely generated as an -algebra by the power-sums

Hence there is a unique -algebra homomorphism

In general we write for , for any . With this notation goes the convention that stands for the sum of the variables, so we have and hence for all . Note that a plethystic substitution like , such as we have on the right-hand side in Equation3, yields again a symmetric function.

There is a simple direct characterization of the transformed Macdonald polynomials .

#### Proposition 2.1.1 (Reference24, Proposition 2.6).

The satisfy

(1)

,

(2)

, and

(3)

,

where denotes a Schur function, is the partition conjugate to , and the ordering is the dominance partial order on partitions of . These conditions characterize uniquely.

We set , where is the Kostka-Macdonald coefficient defined in Reference40, VI, eq. (8.11). This is then related to the transformed Macdonald polynomials by

It is known that has degree at most in , so the positivity conjecture Equation2 from the introduction can be equivalently formulated in terms of .

We have .

### 2.2. The n factorial $n!$ and graded character conjectures

Let

be the polynomial ring in variables. To each -element subset , we associate a polynomial as follows. Let be the elements of listed in some fixed order. Then we define

If is a partition of , its diagram is the set

(Note that in our definition the rows and columns of the diagram are indexed starting with zero.) In the case where is the diagram of a partition, we abbreviate

The polynomial is a kind of bivariate analog of the Vandermonde determinant , which occurs as the special case .

Given a partition of , we denote by

the space spanned by all the iterated partial derivatives of . In Reference15, Garsia and the author proposed the following conjecture, which we will prove as a consequence of Proposition 3.7.3 and Theorem 3.1.

#### Conjecture 2.2.1 ( n factorial $n!$ conjecture).

The dimension of is equal to .

The conjecture arose as part of a stronger conjecture relating the Kostka-Macdonald coefficients to the character of as a doubly graded -module. The symmetric group acts by -algebra automorphisms of permuting the variables:

The ring is doubly graded, by degree in the and variables respectively, and the action respects the grading. Clearly is -alternating, i.e., we have for all , where is the sign character. Note that is also doubly homogeneous, of -degree and -degree . It follows that the space is -invariant and has a double grading

by -invariant subspaces .

We write for the character of an -module , and denote the irreducible characters by , with the usual indexing by partitions of . The following conjecture implies the Macdonald positivity conjecture.

#### Conjecture 2.2.2 (Reference15).

We have

Macdonald had shown that is equal to , the degree of the irreducible character , or the number of standard Young tableaux of shape . Conjecture 2.2.2 therefore implies that affords the regular representation of . In particular, it implies the conjecture.

In Reference24 the author showed that Conjecture 2.2.2 would follow from the Cohen-Macaulay property of . We summarize the argument proving Conjecture 2.2.2 in §3.9, after the relevant geometric results have been established.

## 3. The isospectral Hilbert scheme

### 3.1. Preliminaries

In this section we define the isospectral Hilbert scheme , and deduce our main theorem, Theorem 3.13.8). We also define the Hilbert scheme and the nested Hilbert scheme , and develop some basic properties of these various schemes in preparation for the proof of the main theorem.

The main technical device used in the proof of Theorem 3.1 is a theorem on certain subspace arrangements called polygraphs, Theorem 4.1. The proof of the latter theorem is lengthy and logically distinct from the geometric reasoning leading from there to Theorem 3.1. For these reasons we have deferred Theorem 4.1 and its proof to the separate §4.

Throughout this section we work in the category of schemes of finite type over the field of complex numbers, . All the specific schemes we consider are quasiprojective over . We use classical geometric language, describing open and closed subsets of schemes, and morphisms between reduced schemes, in terms of closed points. A variety is a reduced and irreducible scheme.

Every locally free coherent sheaf of rank on a scheme of finite type over is isomorphic to the sheaf of sections of an algebraic vector bundle of rank over . For notational purposes, we identify the vector bundle with the sheaf and write for the fiber of at a closed point . In sheaf-theoretic terms, the fiber is given by .

A scheme is Cohen-Macaulay or Gorenstein if its local ring at every point is a Cohen-Macaulay or Gorenstein local ring, respectively. For either condition it suffices that it holds at closed points . At the end of the section, in §3.10, we provide a brief summary of the facts we need from duality theory and the theory of Cohen-Macaulay and Gorenstein schemes.

### 3.2. The schemes upper H Subscript n $H_{n}$ and upper X Subscript n $X_{n}$

Let be the coordinate ring of the affine plane . By definition, closed subschemes are in one-to-one correspondence with ideals . The subscheme is finite if and only if has Krull dimension zero, or finite dimension as a vector space over . In this case, the length of is defined to be .

The Hilbert scheme parametrizes finite closed subschemes of length . The scheme structure of and the precise sense in which it parametrizes the subschemes are defined by a universal property, which characterizes up to unique isomorphism. The universal property is actually a property of together with a closed subscheme , called the universal family.

#### Proposition 3.2.1

There exist schemes and enjoying the following properties, which characterize them up to unique isomorphism:

(1)

is flat and finite of degree over , and

(2)

if is a closed subscheme, flat and finite of degree over a scheme , then there is a unique morphism giving a commutative fiber product diagram

that is, the flat family over is the pullback through of the universal family .

#### Proof.

The Hilbert scheme of points in the projective plane exists as a special case of Grothendieck’s construction in Reference21, with a universal family having the analogous universal property. We identify as usual with an open subset of , the complement of the projective line “at infinity”.

The projection of onto is a closed subset of . Its complement is clearly the largest subset such that the restriction of to is contained in . The required universal property of and now follows immediately from that of and .

To see how parametrizes finite closed subschemes of length , note that the latter are exactly the families in Proposition 3.2.1 for . By the universal property they correspond one-to-one with the closed points of , in such a way that the fiber of the universal family over the point corresponding to is itself. For notational purposes we will identify the closed points of with ideals satisfying , rather than with the corresponding subschemes .

We have the following fundamental theorem of Fogarty Reference14.

#### Proposition 3.2.2

The Hilbert scheme is a nonsingular, irreducible variety over of dimension .

The generic examples of finite closed subschemes of length are the reduced subschemes consisting of distinct points. They form an open subset of , and the irreducibility aspect of Fogarty’s theorem means that this open set is dense.

The most special closed subschemes in a certain sense are those defined by monomial ideals. If is a monomial ideal, then the standard monomials form a basis of . If , the exponents of the standard monomials form the diagram of a partition of , and conversely. We use the partition to index the corresponding monomial ideal, denoting it by . Note that for all , so the subscheme is concentrated at the origin , and owes its length entirely to its nonreduced scheme structure.

The algebraic torus

acts on as the group of invertible diagonal matrices. The monomial ideals are the torus invariant ideals, and thus they are the fixed points of the induced action of on the Hilbert scheme. Every ideal has a monomial ideal in the closure of its -orbit (Reference23, Lemma 2.3).

We write for the coordinates on the -th factor in the Cartesian product , so we have , where . The symmetric group acts on by permuting the factors. In coordinates, this corresponds to the action of on given in Equation13. We can identify with the variety

of unordered -tuples, or -element multisets, of points in .

#### Proposition 3.2.3 (Reference23, Proposition 2.2).

For , let be the multiset of points of , counting each point with multiplicity equal to the length of the local ring . Then the map is a projective morphism (called the Chow morphism).

#### Definition 3.2.4

The isospectral Hilbert scheme is the reduced fiber product

that is, the reduced closed subscheme of whose closed points are the tuples satisfying .

We will continue to refer to the morphisms , and in diagram Equation18 by those names in what follows.

For each , the operators , of multiplication by are commuting endomorphisms of the -dimensional vector space . As such, they have a well-defined joint spectrum, a multiset of pairs of eigenvalues determined by the identity

On the local ring at a point , the operators , have the sole joint eigenvalue , with multiplicity equal to the length of . Hence is equal as a multiset to the joint spectrum of and . This is the motivation for the term isospectral.

The action of on induces a compatible action of on by automorphisms of as a scheme over . Explicitly, for we have .

We caution the reader that the scheme-theoretic fiber product in Equation18 is not reduced, even for . For every invariant polynomial , the global regular function

on vanishes on . By definition these equations generate the ideal sheaf of the scheme-theoretic fiber product. They cut out set-theoretically, but not as a reduced subscheme. The full ideal sheaf defining as a reduced scheme must necessarily have a complicated local description, since it is a consequence of Theorem 3.1 and Proposition 3.7.3, below, that generators for all the ideals in §3.7, eq. Equation36, are implicit in the local ideals of at the distinguished points lying over the torus-fixed points .

### 3.3. Elementary properties of upper X Subscript n $X_{n}$

We now develop some elementary facts about the isospectral Hilbert scheme . The first of these is its product structure, which allows us to reduce local questions on to questions about for , in a neighborhood of any point whose corresponding multiset is not of the form .

#### Lemma 3.3.1

Let and be positive integers with . Suppose is an open set consisting of points where no coincides with any . Then, identifying with , the preimage of in is isomorphic as a scheme over to the preimage of in .

#### Proof.

Let be the universal family over . The fiber of over a point is the disjoint union of closed subschemes and in of lengths and , respectively, with and . Hence over , is the disjoint union of flat families , of degrees and . By the universal property, we get induced morphisms , and . The equations , imply that factors through a morphism of schemes over .

Conversely, on , the pullbacks of the universal families from and are disjoint and their union is a flat family of degree . By the universal property there is an induced morphism , which factors through a morphism of schemes over .

By construction, the universal families on and pull back to themselves via and , respectively. This implies that is a morphism of schemes over and is a morphism of schemes over . Since they are also morphisms of schemes over , we have and . Hence and induce mutually inverse isomorphisms .

#### Proposition 3.3.2

The isospectral Hilbert scheme is irreducible, of dimension .

#### Proof.

Let be the preimage in of the open set consisting of points where the are all distinct. It follows from Lemma 3.3.1 that restricts to an isomorphism , so is irreducible. We are to show that is dense in .

Let be a closed point of , which we want to show belongs to the closure of . If with the not all equal, then by Lemma 3.3.1 there is a neighborhood of in isomorphic to an open set in for some . The result then follows by induction, since we may assume and irreducible. If all the are equal, then is the unique point of lying over . Since is finite, is closed. But is dense in , so . Therefore contains a point lying over , which must be .

#### Proposition 3.3.3

The closed subset in has dimension .

#### Proof.

It follows from the cell decomposition of Ellingsrud and Strømme Reference11 that the closed locus in consisting of points with supported on the -axis in is the union of locally closed affine cells of dimension . The subset is equal to and is finite.

The product structure of is inherited in a certain sense by , but its description in terms of is more transparent. As a consequence, passage to is sometimes handy for proving results purely about . The following lemma is an example of this. We remark that one can show by a more careful analysis that locus in the lemma is in fact irreducible.

#### Lemma 3.3.4

Let be the closed subset of consisting of ideals for which contains some point with multiplicity at least . Then has codimension , and has a unique irreducible component of maximal dimension.

#### Proof.

By symmetry among the points of we see that , where is the locus in defined by the equations . It follows from Lemma 3.3.1 that is isomorphic to an open set in , where is the closed subset in . As a reduced subscheme of , the latter is isomorphic to , where is the zero fiber in , the factor accounting for the choice of .

By a theorem of Briançon Reference5, is irreducible of dimension , so is irreducible of dimension . Since and is finite, the result follows by descending induction on , starting with .

### 3.4. Blowup construction of upper H Subscript n $H_{n}$ and upper X Subscript n $X_{n}$

Let be the space of -alternating elements, that is, polynomials such that for all , where is the sign character. To describe more precisely, we note that is the image of the alternation operator

If is an -element subset of , then the determinant defined in Equation9 can also be written

where . For a monomial whose exponent pairs are not all distinct, we have . From this it is easy to see that the set of all elements is a basis of . Another way to see this is to identify with the -th exterior power of the polynomial ring in two variables . Then the basis elements are identified with the wedge products of monomials in .

For , let be the space spanned by all products of elements of . We set . Note that and hence every is a -submodule of , so we have for all , , including or .

#### Proposition 3.4.1 (Reference23, Proposition 2.6).

The Hilbert scheme is isomorphic as a scheme projective over to , where is the graded -algebra .

#### Proposition 3.4.2

The isospectral Hilbert scheme is isomorphic as a scheme over to the blowup of at the ideal generated by the alternating polynomials.

#### Proof.

Set . By definition the blowup of at is , where is the Rees algebra. The ring is a homogeneous subring of in an obvious way, and since generates as a -module, we have , that is, for some homogeneous ideal . In geometric terms, using Proposition 3.4.1 and the fact that , this says that is a closed subscheme of the scheme-theoretic fiber product . Since is reduced, it follows that is a closed subscheme of . By Proposition 3.3.2, is irreducible, and since both and have dimension , it follows that .

In the context of either or we will always write for the -th tensor power of the ample line bundle induced by the representation of as or as . It is immediate from the proof of Proposition 3.4.2 that .

In full analogy to the situation for the Plücker embedding of a Grassmann variety, there is an intrinsic description of as the highest exterior power of the tautological vector bundle whose fiber at a point is . Let

be the projection of the universal family on the Hilbert scheme. Since is an affine morphism, we have , where is the sheaf of -algebras

The fact that is flat and finite of degree over means that is a locally free sheaf of -modules of rank . Its associated vector bundle is the tautological bundle.

#### Proposition 3.4.3 (Reference23, Proposition 2.12).

We have an isomorphism of line bundles on .

### 3.5. Nested Hilbert schemes

The proof of our main theorem will be by induction on . For the inductive step we interpolate between and using the nested Hilbert scheme.

#### Definition 3.5.1

The nested Hilbert scheme is the reduced closed subscheme

The analog of Fogarty’s theorem (Proposition 3.2.2) for the nested Hilbert scheme is the following result of Tikhomirov, whose proof can be found in Reference8.

#### Proposition 3.5.2

The nested Hilbert scheme is nonsingular and irreducible, of dimension .

As with , the nested Hilbert scheme is an open set in a projective nested Hilbert scheme . Clearly, is the preimage of under the projection . Hence the projection is a projective morphism.

If is a point of , then is an element sub-multiset of . In symbols, if , then for a distinguished last point . The -invariant polynomials in the coordinates of the points are global regular functions on , pulled back via the projection on . Similarly, the -invariant polynomials in are regular functions pulled back from . It follows that the coordinates of the distinguished point are regular functions, since , and similarly for . Thus we have a morphism

such that both the maps and induced by the Chow morphisms composed with the projections on and factor through .

The distinguished point belongs to , and given , every point of occurs as for some choice of . Therefore the image of the morphism

sending to is precisely the universal family over . For clarity, let us point out that by the definition of , we have , at least set-theoretically. In fact, is reduced and hence coincides as a reduced closed subscheme with this subset of . This is true because is flat over the variety and generically reduced (see also the proof of Proposition 3.7.2 below).

The following proposition, in conjunction with Lemma 3.3.4, provides dimension estimates needed for the calculation of the canonical line bundle on in §3.6 and the proof of the main theorem in §3.8.

#### Proposition 3.5.3

Let be the dimension of the fiber of the morphism in Equation27 over a point , and let be the multiplicity of in . Then and satisfy the inequality

#### Proof.

Recall that the socle of an Artin local ring is the ideal consisting of elements annihilated by the maximal ideal . If is an algebra over a field , with , then every linear subspace of the socle is an ideal, and conversely every ideal in of length is a one-dimensional subspace of the socle. The possible ideals for the given are the length ideals in the Artin local -algebra , where . The fiber of is therefore the projective space , and we have .

First consider the maximum possible dimension of any fiber of . Since both and are projective over , the morphism is projective and its fiber dimension is upper semicontinuous. Since every point of has a monomial ideal in the closure of its -orbit, and since is finite over , every point of must have a pair in the closure of its orbit. The fiber dimension is therefore maximized at some such point. The socle of has dimension equal to the number of corners of the diagram of . If this number is , we clearly have . This implies that for every Artin local -algebra generated by two elements, the socle dimension and the length of satisfy .

Returning to the original problem, is an Artin local -algebra of length generated by two elements, with socle dimension , so Equation28 follows.

We now introduce the nested version of the isospectral Hilbert scheme. It literally plays a pivotal role in the proof of the main theorem by induction on : we transfer the Gorenstein property from to the nested scheme by pulling back, and from there to by pushing forward.

#### Definition 3.5.4

The nested isospectral Hilbert scheme is the reduced fiber product .

There is an alternative formulation of the definition, which is useful to keep in mind for the next two results. Namely, can be identified with the reduced fiber product in the diagram

that is, the reduced closed subscheme of consisting of tuples such that and is the distinguished point. To see that this agrees with the definition, note that a point of is given by the data , and that these data determine the distinguished point . We obtain the alternative description by identifying with the graph in of the morphism sending to .

We have the following nested analogs of Lemma 3.3.1 and Proposition 3.3.3. The analog of Proposition 3.3.2 also holds, i.e., is irreducible. We do not prove this here, as it will follow automatically as part of our induction (see the observations following diagram Equation52 in §3.8).

#### Lemma 3.5.5

Let and be as in Lemma 3.3.1. Then the preimage of in is isomorphic as a scheme over to the preimage of in .

#### Proof.

Lemma 3.3.1 gives us isomorphisms on the preimage of between and , and between and .

We can identify with the closed subset of consisting of points where are the same in both factors, and contains . On the preimage of , under the isomorphisms above, this corresponds to the closed subset of where , and are the same in both factors and contains . The latter can be identified with .

#### Proposition 3.5.6

The closed subset