Hilbert schemes, polygraphs and the Macdonald positivity conjecture

Abstract

We study the isospectral Hilbert scheme $X_{n}$, defined as the reduced fiber product of $(\mathbb{C}^{2})^{n}$ with the Hilbert scheme $H_{n}$ of points in the plane $\mathbb{C}^{2}$, over the symmetric power $S^{n}\mathbb{C}^{2} = (\mathbb{C}^{2})^{n}/S_{n}$. By a theorem of Fogarty, $H_{n}$ is smooth. We prove that $X_{n}$ is normal, Cohen-Macaulay and Gorenstein, and hence flat over $H_{n}$. We derive two important consequences.

(1) We prove the strong form of the $n!$ conjecture of Garsia and the author, giving a representation-theoretic interpretation of the Kostka-Macdonald coefficients $K_{\lambda \mu }(q,t)$. This establishes the Macdonald positivity conjecture, namely that $K_{\lambda \mu }(q,t)\in {\mathbb{N}} [q,t]$.

(2) We show that the Hilbert scheme $H_{n}$ is isomorphic to the $G$-Hilbert scheme $(\mathbb{C}^{2})^{n}{/\!\!/}S_n$ of Nakamura, in such a way that $X_{n}$ is identified with the universal family over $({\mathbb{C}}^2)^n{/\!\!/}S_n$. From this point of view, $K_{\lambda \mu }(q,t)$ describes the fiber of a character sheaf $C_{\lambda }$ at a torus-fixed point of $({\mathbb{C}}^2)^n{/\!\!/}S_n$ corresponding to $\mu$.

The proofs rely on a study of certain subspace arrangements $Z(n,l)\subseteq (\mathbb{C}^{2})^{n+l}$, called polygraphs, whose coordinate rings $R(n,l)$ carry geometric information about $X_{n}$. The key result is that $R(n,l)$ is a free module over the polynomial ring in one set of coordinates on $(\mathbb{C}^{2})^{n}$. This is proven by an intricate inductive argument based on elementary commutative algebra.

1. Introduction

The Hilbert scheme of points in the plane $H_{n} = \operatorname {Hilb}^{n}(\mathbb{C}^{2})$ is an algebraic variety which parametrizes finite subschemes $S$ of length $n$ in $\mathbb{C}^{2}$. To each such subscheme $S$ corresponds an $n$-element multiset, or unordered $n$-tuple with possible repetitions, $\sigma (S) = \lBrack P_{1},\ldots ,P_{n} \rBrack$ of points in $\mathbb{C}^{2}$, where the $P_{i}$ are the points of $S$, repeated with appropriate multiplicities. There is a variety $X_{n}$, finite over $H_{n}$, whose fiber over the point of $H_{n}$ corresponding to $S$ consists of all ordered $n$-tuples $(P_{1},\ldots ,P_{n})\in (\mathbb{C}^{2})^{n}$ whose underlying multiset is $\sigma (S)$. We call $X_{n}$ the isospectral Hilbert scheme.

By a theorem of Fogarty 14, the Hilbert scheme $H_{n}$ is irreducible and nonsingular. The geometry of $X_{n}$ is more complicated, but also very special. Our main geometric result, Theorem 3.1, is that $X_{n}$ is normal, Cohen-Macaulay and Gorenstein.

Earlier investigations by the author 24 unearthed indications of a far-reaching correspondence between the geometry and sheaf cohomology of $H_{n}$ and $X_{n}$ on the one hand, and the theory of Macdonald polynomials on the other. The Macdonald polynomials

$$\begin{equation} P_{\mu }(x;q,t) {\tag{1}} \end{equation}$$

are a basis of the algebra of symmetric functions in variables $x = x_{1},x_{2},\ldots \,$, with coefficients in the field $\mathbb{Q}(q,t)$ of rational functions in two parameters $q$ and $t$. They were introduced in 1988 by Macdonald 39 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 40). 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 9 and other discoveries relating Macdonald polynomials to the representation theory of quantum groups 13 and affine Hecke algebras 323341, the Calogero-Sutherland model in particle physics 35, and combinatorial conjectures on diagonal harmonics 31622.

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 $K_{\lambda \mu }(q,t)$, known as Kostka-Macdonald coefficients. As defined, they are rational functions of $q$ and $t$, but conjecturally they are polynomials in $q$ and $t$ with nonnegative integer coefficients:

$$\begin{equation} K_{\lambda \mu }(q,t)\in \mathbb{N}[q,t]. {\tag{2}} \end{equation}$$

The positivity conjecture has remained open since Macdonald formulated it at the time of his original discovery. For $q=0$ it reduces to the positivity theorem for $t$-Kostka coefficients, which has important algebraic, geometric and combinatorial interpretations 7101727313436373845. Only recently have several authors independently shown that the Kostka-Macdonald coefficients are polynomials, $K_{\lambda \mu }(q,t)\in \mathbb{Z}[q,t]$, but these results do not establish the positivity 1819323344.

In 15, Garsia and the author conjectured an interpretation of the Kostka-Macdonald coefficients $K_{\lambda \mu }(q,t)$ as graded character multiplicities for certain doubly graded $S_{n}$-modules $D_{\mu }$. The module $D_{\mu }$ is the space of polynomials in $2n$ 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 $K_{\lambda \mu }(q,t)$, that for each partition $\mu$ of $n$, the dimension of $D_{\mu }$ is equal to $n!$. This seemingly elementary assertion has come to be known as the $n!$ conjecture.

It develops that these conjectures are closely tied to the geometry of the isospectral Hilbert scheme. Specifically, in 24 we were able to show that the Cohen-Macaulay property of $X_{n}$ is equivalent to the $n!$ conjecture. We further showed that the Cohen-Macaulay property of $X_{n}$ implies the stronger conjecture interpreting $K_{\lambda \mu }(q,t)$ as a graded character multiplicity for $D_{\mu }$. 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 $H_{n}$ is equal to the $G$-Hilbert scheme $V\mathord{\sslash }G$ of Ito and Nakamura 28, for the case $V = (\mathbb{C}^{2})^{n}$, $G = S_{n}$. The $G$-Hilbert scheme is of interest in connection with the generalized McKay correspondence, which says that if $V$ is a complex vector space, $G$ is a finite subgroup of $\operatorname {SL}(V)$ and $Y\rightarrow V/G$ is a so-called crepant resolution of singularities, then the sum of the Betti numbers of $Y$ equals the number of conjugacy classes of $G$. In many interesting cases 642, the $G$-Hilbert scheme turns out to be a crepant resolution and an instance of the McKay correspondence. By our main theorem, this holds for $G = S_{n}$, $V = (\mathbb{C}^{2})^{n}$.

We wish to say a little at this point about how the discoveries presented here came about. It has long been known 2745 that the $t$-Kostka coefficients $K_{\lambda \mu }(t) = K_{\lambda \mu }(0,t)$ are graded character multiplicities for the cohomology rings of Springer fibers. Garsia and Procesi 17 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 $K_{\lambda \mu }(t)$ 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 $n!$ conjecture. Based on Garsia and Procesi’s experience, we initially expected that the $n!$ conjecture itself would be easy to prove and that the difficulties would lie in the identification of $K_{\lambda \mu }(q,t)$ as the graded character multiplicity. To our surprise, however, the $n!$ conjecture stubbornly resisted elementary attack.

In the spring of 1992, we discussed our efforts on the $n!$ conjecture with Procesi, along with another related conjecture we had stumbled upon in the course of our work. The modules involved in the $n!$ conjecture are quotients of the ring $R_{n}$ of coinvariants for the action of $S_{n}$ on the polynomial ring in $2n$ variables. This ring $R_{n}$ is isomorphic to the space of diagonal harmonics. Computations suggested that its dimension should be $({n+1})^{n-1}$ 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 1622). Procesi suggested that the Hilbert scheme $H_{n}$ and what we now call the isospectral Hilbert scheme $X_{n}$ should be relevant to the determination of the dimension and character of $R_{n}$. Specifically, he observed that there is a natural map from $R_{n}$ to the ring of global functions on the scheme-theoretic fiber in $X_{n}$ over the origin in $S^{n}\mathbb{C}^{2}$. With luck, this map might be an isomorphism, and—as we are now able to confirm—$X_{n}$ might be flat over $H_{n}$, so that its structure sheaf would push down to a vector bundle on $H_{n}$. Then $R_{n}$ would coincide with the space of global sections of this vector bundle over the zero-fiber in $H_{n}$, and it might be possible to compute its character using the Atiyah-Bott Lefschetz formula.

The connection between $X_{n}$ and the $n!$ 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 $n!$ conjecture should be the fibers of Procesi’s vector bundle at distinguished torus-fixed points in $H_{n}$, a fact which we prove in §3.7. These considerations ultimately led to a conjectured formula for the character of $R_{n}$ in terms of Macdonald polynomials. This formula turned out to be correct up to the limit of practical computation ($n\leq 7$). Furthermore, Garsia and the author were able to show in 16 that the series of combinatorial conjectures in 22 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 $X_{n}$ should ultimately lead to a proof of the $n!$ and Macdonald positivity conjectures, as is borne out here. By now the reader should expect the geometric study of $X_{n}$ also to yield a proof of the character formula for diagonal harmonics and the $({n+1})^{n-1}$ 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, $n!$ 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 $n!$ conjecture. In §3.9 we review the proof from 24 that the main theorem implies the conjecture of Garsia and the author on the character of the space $D_{\mu }$, 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 $G$-Hilbert schemes, along with related conjectures and open problems.

2. The $n!$ and Macdonald positivity conjectures

2.1. Macdonald polynomials

We work with the transformed integral forms $\tilde{H}_{\mu }(x;q,t)$ of the Macdonald polynomials, indexed by integer partitions $\mu$, and homogeneous of degree $n = |\mu |$. These are defined as in 24, eq. (2.18), to be

$$\begin{equation} \tilde{H}_{\mu }(x;q,t) = t^{n(\mu )}J_{\mu }[X/(1-t^{-1});q,t^{-1}], {\tag{3}} \end{equation}$$

where $J_{\mu }$ denotes Macdonald’s integral form as in 40, VI, eq. (8.3), and $n(\mu )$ is the partition statistic

$$\begin{equation} n(\mu ) = \sum _{i}(i-1)\mu _{i} {\tag{4}} \end{equation}$$

(not to be confused with $n = |\mu |$).

The square brackets in 3 stand for plethystic substitution. We pause briefly to review the definition of this operation (see 24 for a fuller discussion). Let $\mathbb{F}[[x]]$ be the algebra of formal series over the coefficient field $\mathbb{F}= \mathbb{Q}(q,t)$, in variables $x = x_{1},x_{2},\ldots \;$. For any $A\in \mathbb{F}[[x]]$, we denote by $p_{k}[A]$ the result of replacing each indeterminate in $A$ by its $k$-th power. This includes the indeterminates $q$ and $t$ as well as the variables $x_{i}$. The algebra of symmetric functions $\Lambda _{\mathbb{F}}$ is freely generated as an $\mathbb{F}$-algebra by the power-sums

$$\begin{equation} p_{k}(x) = x_{1}^{k}+x_{2}^{k}+\cdots . {\tag{5}} \end{equation}$$

Hence there is a unique $\mathbb{F}$-algebra homomorphism

$$\begin{equation} \operatorname {ev}_{A}\colon \Lambda _{\mathbb{F}}\rightarrow \mathbb{F}[[x]]\quad \text{defined by}\quad p_{k}(x)\mapsto p_{k}[A]. {\tag{6}} \end{equation}$$

In general we write $f[A]$ for $\operatorname {ev}_{A}(f)$, for any $f\in \Lambda _{\mathbb{F}}$. With this notation goes the convention that $X$ stands for the sum $X = x_{1}+x_{2}+\cdots$ of the variables, so we have $p_{k}[X] = p_{k}(x)$ and hence $f[X] = f(x)$ for all $f$. Note that a plethystic substitution like $f\mapsto f[X/(1-t^{-1})]$, such as we have on the right-hand side in 3, yields again a symmetric function.

There is a simple direct characterization of the transformed Macdonald polynomials $\tilde{H}_{\mu }$.

Proposition 2.1.1 (24, Proposition 2.6).

The $\tilde{H}_{\mu }(x;q,t)$ satisfy

(1)

$\tilde{H}_{\mu }(x;q,t)\in \mathbb{Q}(q,t)\{ s_{\lambda }[X/(1-q)] : \lambda \geq \mu \}$,

(2)

$\tilde{H}_{\mu }(x;q,t)\in \mathbb{Q}(q,t)\{ s_{\lambda }[X/(1-t)] : \lambda \geq \mu ' \}$, and

(3)

$\tilde{H}_{\mu }[1;q,t] = 1$,

where $s_{\lambda }(x)$ denotes a Schur function, $\mu '$ is the partition conjugate to $\mu$, and the ordering is the dominance partial order on partitions of $n = |\mu |$. These conditions characterize $\tilde{H}_{\mu }(x;q,t)$ uniquely.

We set $\tilde{K}_{\lambda \mu }(q,t) = t^{n(\mu )}K_{\lambda \mu }(q,t^{-1})$, where $K_{\lambda \mu }(q,t)$ is the Kostka-Macdonald coefficient defined in 40, VI, eq. (8.11). This is then related to the transformed Macdonald polynomials by

$$\begin{equation} \tilde{H}_{\mu }(x;q,t) = \sum _{\lambda } \tilde{K}_{\lambda \mu }(q,t) s_{\lambda }(x). {\tag{7}} \end{equation}$$

It is known that $K_{\lambda \mu }(q,t)$ has degree at most $n(\mu )$ in $t$, so the positivity conjecture 2 from the introduction can be equivalently formulated in terms of $\tilde{K}_{\lambda \mu }$.

Conjecture 2.1.2 (Macdonald positivity conjecture).

We have $\tilde{K}_{\lambda \mu }(q,t)\!\in \! \mathbb{N}[q,t]$.

2.2. The $n!$ and graded character conjectures

Let

$$\begin{equation} \mathbb{C}[\mathbf{x},\mathbf{y}] = \mathbb{C}[x_{1},y_{1}, \ldots , x_{n},y_{n}] \tag{8} \end{equation}$$

be the polynomial ring in $2n$ variables. To each $n$-element subset $D\subseteq \mathbb{N}\times \mathbb{N}$, we associate a polynomial $\Delta _{D}\in \mathbb{C}[\mathbf{x},\mathbf{y}]$ as follows. Let $(p_{1},q_{1}),\ldots ,(p_{n},q_{n})$ be the elements of $D$ listed in some fixed order. Then we define

$$\begin{equation} \Delta _{D} = \det \left(x_{i}^{p_{j}} y_{i}^{q_{j}} \right)_{1\leq i,j\leq n}. {\tag{9}} \end{equation}$$

If $\mu$ is a partition of $n$, its diagram is the set

$$\begin{equation} D(\mu ) = \{(i,j)\colon j<\mu _{i+1} \}\subseteq \mathbb{N}\times \mathbb{N}. {\tag{10}} \end{equation}$$

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

$$\begin{equation} \Delta _{\mu } = \Delta _{D(\mu )}. {\tag{11}} \end{equation}$$

The polynomial $\Delta _{\mu }(\mathbf{x},\mathbf{y})$ is a kind of bivariate analog of the Vandermonde determinant $\Delta (\mathbf{x})$, which occurs as the special case $\mu = (1^{n})$.

Given a partition $\mu$ of $n$, we denote by

$$\begin{equation} D_{\mu } = \mathbb{C}[\partial \mathbf{x},\partial \mathbf{y}]\Delta _{\mu } {\tag{12}} \end{equation}$$

the space spanned by all the iterated partial derivatives of $\Delta _{\mu }$. In 15, 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 ($n!$ conjecture).

The dimension of $D_{\mu }$ is equal to $n!$.

The $n!$ conjecture arose as part of a stronger conjecture relating the Kostka-Macdonald coefficients to the character of $D_{\mu }$ as a doubly graded $S_{n}$-module. The symmetric group $S_{n}$ acts by $\mathbb{C}$-algebra automorphisms of $\mathbb{C}[\mathbf{x},\mathbf{y}]$ permuting the variables:

$$\begin{equation} wx_{i} = x_{w(i)},\; wy_{i} = y_{w(i)}\quad \text{for}\quad w\in S_{n}. {\tag{13}} \end{equation}$$

The ring $\mathbb{C}[\mathbf{x},\mathbf{y}] = \bigoplus _{r,s} \mathbb{C}[\mathbf{x},\mathbf{y}]_{r,s}$ is doubly graded, by degree in the $\mathbf{x}$ and $\mathbf{y}$ variables respectively, and the $S_{n}$ action respects the grading. Clearly $\Delta _{\mu }$ is $S_{n}$-alternating, i.e., we have $w\Delta _{\mu } = \epsilon (w)\Delta _{\mu }$ for all $w\in S_{n}$, where $\epsilon$ is the sign character. Note that $\Delta _{\mu }$ is also doubly homogeneous, of $x$-degree $n(\mu )$ and $y$-degree $n(\mu ')$. It follows that the space $D_{\mu }$ is $S_{n}$-invariant and has a double grading

$$\begin{equation} D_{\mu } = \bigoplus _{r,s} (D_{\mu })_{r,s} {\tag{14}} \end{equation}$$

by $S_{n}$-invariant subspaces $(D_{\mu })_{r,s} = D_{\mu }\cap \mathbb{C}[\mathbf{x},\mathbf{y}]_{r,s}$.

We write $\operatorname {ch}V$ for the character of an $S_{n}$-module $V$, and denote the irreducible $S_{n}$ characters by $\chi ^{\lambda }$, with the usual indexing by partitions $\lambda$ of $n$. The following conjecture implies the Macdonald positivity conjecture.

Conjecture 2.2.2 (15).

We have

$$\begin{equation} \tilde{K}_{\lambda \mu }(q,t) = \sum _{r,s}t^{r}q^{s}\langle \chi ^{\lambda }, \operatorname {ch}(D_{\mu })_{r,s} \rangle . {\tag{15}} \end{equation}$$

Macdonald had shown that $K_{\lambda \mu }(1,1)$ is equal to $\chi ^{\lambda }(1)$, the degree of the irreducible $S_{n}$ character $\chi ^{\lambda }$, or the number of standard Young tableaux of shape $\lambda$. Conjecture 2.2.2 therefore implies that $D_{\mu }$ affords the regular representation of $S_{n}$. In particular, it implies the $n!$ conjecture.

In 24 the author showed that Conjecture 2.2.2 would follow from the Cohen-Macaulay property of $X_{n}$. 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 $X_{n}$, and deduce our main theorem, Theorem 3.1 (§3.8). We also define the Hilbert scheme $H_{n}$ and the nested Hilbert scheme $H_{n-1,n}$, 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, $\mathbb{C}$. All the specific schemes we consider are quasiprojective over $\mathbb{C}$. 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 $B$ of rank $n$ on a scheme $X$ of finite type over $\mathbb{C}$ is isomorphic to the sheaf of sections of an algebraic vector bundle of rank $n$ over $X$. For notational purposes, we identify the vector bundle with the sheaf $B$ and write $B(x)$ for the fiber of $B$ at a closed point $x\in X$. In sheaf-theoretic terms, the fiber is given by $B(x) = B\otimes _{\mathcal{O}_{X}} (\mathcal{O}_{X,x}/x)$.

A scheme $X$ is Cohen-Macaulay or Gorenstein if its local ring $\mathcal{O}_{X,x}$ at every point is a Cohen-Macaulay or Gorenstein local ring, respectively. For either condition it suffices that it holds at closed points $x$. 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 $H_{n}$ and $X_{n}$

Let $R = \mathbb{C}[x,y]$ be the coordinate ring of the affine plane $\mathbb{C}^{2}$. By definition, closed subschemes $S\subseteq \mathbb{C}^{2}$ are in one-to-one correspondence with ideals $I\subseteq R$. The subscheme $S=V(I)$ is finite if and only if $R/I$ has Krull dimension zero, or finite dimension as a vector space over $\mathbb{C}$. In this case, the length of $S$ is defined to be $\dim _{\mathbb{C}}R/I$.

The Hilbert scheme $H_{n} = \operatorname {Hilb}^{n}(\mathbb{C}^{2})$ parametrizes finite closed subschemes $S\subseteq \mathbb{C}^{2}$ of length $n$. The scheme structure of $H_{n}$ and the precise sense in which it parametrizes the subschemes $S$ are defined by a universal property, which characterizes $H_{n}$ up to unique isomorphism. The universal property is actually a property of $H_{n}$ together with a closed subscheme $F\subseteq H_{n}\times \mathbb{C}^{2}$, called the universal family.

Proposition 3.2.1.

There exist schemes $H_{n} = \operatorname {Hilb}^{n}(\mathbb{C}^{2})$ and $F\subseteq H_{n}\times \mathbb{C}^{2}$ enjoying the following properties, which characterize them up to unique isomorphism:

(1)

$F$ is flat and finite of degree $n$ over $H_{n}$, and

(2)

if $Y\subseteq T\times \mathbb{C}^{2}$ is a closed subscheme, flat and finite of degree $n$ over a scheme $T$, then there is a unique morphism $\phi \colon T\rightarrow H_{n}$ giving a commutative fiber product diagram$$\begin{equation*} \begin{CD} Y @>>> T\times \mathbb{C}^{2} @>>> T \\@VVV @VVV @V{\phi }VV\\F @>>> H_{n}\times \mathbb{C}^{2} @>>> H_{n}, \end{CD} \end{equation*}$$

that is, the flat family $Y$ over $T$ is the pullback through $\phi$ of the universal family $F$.

Proof.

The Hilbert scheme $\hat{H} = \operatorname {Hilb}^{n}(\mathbb{P}^{2})$ of points in the projective plane exists as a special case of Grothendieck’s construction in 21, with a universal family $\hat{F}$ having the analogous universal property. We identify $\mathbb{C}^{2}$ as usual with an open subset of $\mathbb{P}^{2}$, the complement of the projective line $Z$ “at infinity”.

The projection of $\hat{F}\cap (\hat{H}\times Z)$ onto $\hat{H}$ is a closed subset of $\hat{H}$. Its complement $H_{n}\subseteq \hat{H}$ is clearly the largest subset such that the restriction $F$ of $\hat{F}$ to $H_{n}$ is contained in $H_{n}\times \mathbb{C}^{2}$. The required universal property of $H_{n}$ and $F$ now follows immediately from that of $\hat{H}$ and $\hat{F}$.

■

To see how $H_{n}$ parametrizes finite closed subschemes $S\subseteq \mathbb{C}^{2}$ of length $n$, note that the latter are exactly the families $Y$ in Proposition 3.2.1 for $T = \operatorname {Spec}\mathbb{C}$. By the universal property they correspond one-to-one with the closed points of $H_{n}$, in such a way that the fiber of the universal family $F$ over the point corresponding to $S$ is $S$ itself. For notational purposes we will identify the closed points of $H_{n}$ with ideals $I \subseteq R$ satisfying $\dim _{\mathbb{C}}R/I = n$, rather than with the corresponding subschemes $S=V(I)$.

We have the following fundamental theorem of Fogarty 14.

Proposition 3.2.2.

The Hilbert scheme $H_{n}$ is a nonsingular, irreducible variety over $\mathbb{C}$ of dimension $2n$.

The generic examples of finite closed subschemes $S\subseteq \mathbb{C}^{2}$ of length $n$ are the reduced subschemes consisting of $n$ distinct points. They form an open subset of $H_{n}$, 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 $I\subseteq R$ is a monomial ideal, then the standard monomials $x^{p}y^{q}\not \in I$ form a basis of $R/I$. If $\dim _{\mathbb{C}}R/I = n$, the exponents $(p,q)$ of the standard monomials form the diagram $D(\mu )$ of a partition $\mu$ of $n$, and conversely. We use the partition $\mu$ to index the corresponding monomial ideal, denoting it by $I_{\mu }$. Note that $\sqrt {\vphantom (}I_{\mu } = (x,y)$ for all $\mu$, so the subscheme $V(I_{\mu })$ is concentrated at the origin $(0,0)\in \mathbb{C}^{2}$, and owes its length entirely to its nonreduced scheme structure.

The algebraic torus

$$\begin{equation} \mathbb{T}^{2} = (\mathbb{C}^{*})^{2} {\tag{16}} \end{equation}$$

acts on $\mathbb{C}^{2}$ as the group of invertible diagonal $2\times 2$ matrices. The monomial ideals $I_{\mu }$ are the torus invariant ideals, and thus they are the fixed points of the induced action of $\mathbb{T}^{2}$ on the Hilbert scheme. Every ideal $I\in H_{n}$ has a monomial ideal in the closure of its $\mathbb{T}^{2}$-orbit (23, Lemma 2.3).

We write $x_{i},y_{i}$ for the coordinates on the $i$-th factor in the Cartesian product $(\mathbb{C}^{2})^{n}$, so we have $(\mathbb{C}^{2})^{n} = \operatorname {Spec}\mathbb{C}[\mathbf{x},\mathbf{y}]$, where