Hilbert schemes, polygraphs and the Macdonald positivity conjecture
By Mark Haiman
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, $\sigma (S) = \llbracket P_{1},\mathinner {\ldotp \ldotp \ldotp },P_{n} \rrbracket$ 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$(P_{1},\mathinner {\ldotp \ldotp \ldotp },P_{n})\in (\mathbb{C}^{2})^{n}$ whose underlying multiset is $\sigma (S)$. We call $X_{n}$ the isospectral Hilbert scheme.
By a theorem of Fogarty Reference14, 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 Reference24 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
are a basis of the algebra of symmetric functions in variables $x = x_{1},x_{2},\mathinner {\ldotp \ldotp \ldotp }\,$, 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 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 $K_{\lambda \mu }(q,t)$, 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:
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 Reference7Reference10Reference17Reference27Reference31Reference34Reference36Reference37Reference38Reference45. 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 Reference18Reference19Reference32Reference33Reference44.
In Reference15, 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 Reference24 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 Reference28, 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 Reference6Reference42, 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 Reference27Reference45 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 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 $K_{\lambda \mu }(t)$ 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 Reference16Reference22). 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 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 $X_{n}$ 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 Reference24 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 Reference24, eq. (2.18), to be
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 $\mathbb{F}[[x]]$ be the algebra of formal series over the coefficient field $\mathbb{F}= \mathbb{Q}(q,t)$, in variables $x = x_{1},x_{2},\mathinner {\ldotp \ldotp \ldotp }\;$. 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
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 Equation3, yields again a symmetric function.
There is a simple direct characterization of the transformed Macdonald polynomials $\tilde{H}_{\mu }$.
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 Reference40, VI, eq. (8.11). This is then related to the transformed Macdonald polynomials by
It is known that $K_{\lambda \mu }(q,t)$ has degree at most $n(\mu )$ in $t$, so the positivity conjecture Equation2 from the introduction can be equivalently formulated in terms of $\tilde{K}_{\lambda \mu }$.
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 $(p_{1},q_{1}),\mathinner {\ldotp \ldotp \ldotp },(p_{n},q_{n})$ be the elements of $D$ listed in some fixed order. Then we define
(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
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 $\mu = (1^{n})$.
the space spanned by all the iterated partial derivatives of $\Delta _{\mu }$. 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 ($n!$ 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:
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
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.
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 Reference24 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 Reference21, 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 Reference14.
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.
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 (Reference23, 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 $\mathbb{C}[\mathbf{x},\mathbf{y}] = \mathbb{C}[x_{1},y_{1},\mathinner {\ldotp \ldotp \ldotp },x_{n},y_{n}]$. The symmetric group $S_{n}$ acts on $(\mathbb{C}^{2})^{n}$ by permuting the factors. In coordinates, this corresponds to the action of $S_{n}$ on $\mathbb{C}[\mathbf{x},\mathbf{y}]$ given in Equation13. We can identify $\operatorname {Spec}\mathbb{C}[\mathbf{x},\mathbf{y}]^{S_{n}}$ with the variety
For $I\in H_{n}$, let $\sigma (I)$ be the multiset of points of $V(I)$, counting each point $P$ with multiplicity equal to the length of the local ring $(R/I)_{P}$. Then the map $\sigma \colon H_{n}\rightarrow S^{n}\mathbb{C}^{2}$ is a projective morphism (called the Chow morphism).
Definition 3.2.4
The isospectral Hilbert scheme$X_{n}$ is the reduced fiber product
that is, the reduced closed subscheme of $H_{n}\times (\mathbb{C}^{2})^{n}$ whose closed points are the tuples $(I,P_{1},\mathinner {\ldotp \ldotp \ldotp },P_{n})$ satisfying $\sigma (I) = \llbracket P_{1},\mathinner {\ldotp \ldotp \ldotp },P_{n} \rrbracket$.
We will continue to refer to the morphisms $\rho$,$\sigma$ and $f$ in diagram Equation18 by those names in what follows.
For each $I\in H_{n}$, the operators $\overline{x}$,$\overline{y}$ of multiplication by $x,y$ are commuting endomorphisms of the $n$-dimensional vector space $R/I$. As such, they have a well-defined joint spectrum, a multiset of pairs of eigenvalues $(x_{1},y_{1}),\mathinner {\ldotp \ldotp \ldotp },(x_{n},y_{n})$ determined by the identity
On the local ring $(R/I)_{P}$ at a point $P = (x_{0},y_{0})$, the operators $\overline{x}$,$\overline{y}$ have the sole joint eigenvalue $(x_{0},y_{0})$, with multiplicity equal to the length of $(R/I)_{P}$. Hence $\sigma (I)$ is equal as a multiset to the joint spectrum of $\overline{x}$ and $\overline{y}$. This is the motivation for the term isospectral.
The action of $S_{n}$ on $(\mathbb{C}^{2})^{n}$ induces a compatible action of $S_{n}$ on $X_{n}$ by automorphisms of $X_{n}$ as a scheme over $H_{n}$. Explicitly, for $w\!\in \! S_{n}$ we have $w(I,P_{1},\mathinner {\ldotp \ldotp \ldotp },P_{n})\! =\! (I,P_{w^{-1}(1)},\mathinner {\ldotp \ldotp \ldotp },P_{w^{-1}(n)})$.
We caution the reader that the scheme-theoretic fiber product in Equation18 is not reduced, even for $n=2$. For every invariant polynomial $g\in \mathbb{C}[\mathbf{x},\mathbf{y}]^{S_{n}}$, the global regular function
on $H_{n}\times (\mathbb{C}^{2})^{n}$ vanishes on $X_{n}$. By definition these equations generate the ideal sheaf of the scheme-theoretic fiber product. They cut out $X_{n}$ set-theoretically, but not as a reduced subscheme. The full ideal sheaf defining $X_{n}$ 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 $J_{\mu }$ in §3.7, eq. Equation36, are implicit in the local ideals of $X_{n}$ at the distinguished points $Q_{\mu }$ lying over the torus-fixed points $I_{\mu }\in H_{n}$.
3.3. Elementary properties of $X_{n}$
We now develop some elementary facts about the isospectral Hilbert scheme $X_{n}$. The first of these is its product structure, which allows us to reduce local questions on $X_{n}$ to questions about $X_{k}$ for $k<n$, in a neighborhood of any point whose corresponding multiset $\llbracket P_{1},\mathinner {\ldotp \ldotp \ldotp }, P_{n} \rrbracket$ is not of the form $\llbracket n\cdot P \rrbracket$.
Lemma 3.3.1
Let $k$ and $l$ be positive integers with $k+l = n$. Suppose $U\subseteq (\mathbb{C}^{2})^{n}$ is an open set consisting of points $(P_{1},\mathinner {\ldotp \ldotp \ldotp },P_{k},Q_{1},\mathinner {\ldotp \ldotp \ldotp },Q_{l})$ where no $P_{i}$ coincides with any $Q_{j}$. Then, identifying $(\mathbb{C}^{2})^{n}$ with $(\mathbb{C}^{2})^{k}\times (\mathbb{C}^{2})^{l}$, the preimage $f^{-1}(U)$ of $U$ in $X_{n}$ is isomorphic as a scheme over $(\mathbb{C}^{2})^{n}$ to the preimage $(f_{k}\times f_{l})^{-1}(U)$ of $U$ in $X_{k}\times X_{l}$.
Proof.
Let $Y = (\rho \times 1_{\mathbb{C}^{2}})^{-1}(F)\subseteq X_{n}\times \mathbb{C}^{2}$ be the universal family over $X_{n}$. The fiber $V(I)$ of $Y$ over a point $(I,P_{1}, \mathinner {\ldotp \ldotp \ldotp }, P_{k}, Q_{1}, \mathinner {\ldotp \ldotp \ldotp }, Q_{l})\in f^{-1}(U)$ is the disjoint union of closed subschemes $V(I_{k})$ and $V(I_{l})$ in $\mathbb{C}^{2}$ of lengths $k$ and $l$, respectively, with $\sigma (I_{k}) = \llbracket P_{1},\mathinner {\ldotp \ldotp \ldotp },P_{k} \rrbracket$ and $\sigma (I_{l}) = \llbracket Q_{1},\mathinner {\ldotp \ldotp \ldotp },Q_{l} \rrbracket$. Hence over $f^{-1}(U)$,$Y$ is the disjoint union of flat families $Y_{k}$,$Y_{l}$ of degrees $k$ and $l$. By the universal property, we get induced morphisms $\phi _{k}\colon f^{-1}(U)\rightarrow H_{k}$,$\phi _{l}\colon f^{-1}(U)\rightarrow H_{l}$ and $\phi _{k}\times \phi _{l}\colon f^{-1}(U)\rightarrow H_{k}\times H_{l}$. The equations $\sigma (I_{k}) = \llbracket P_{1},\mathinner {\ldotp \ldotp \ldotp },P_{k} \rrbracket$,$\sigma (I_{l}) = \llbracket Q_{1},\mathinner {\ldotp \ldotp \ldotp },Q_{l} \rrbracket$ imply that $\phi _{k}\times \phi _{l}$ factors through a morphism $\alpha \colon f^{-1}(U)\rightarrow X_{k}\times X_{l}$ of schemes over $(\mathbb{C}^{2})^{n}$.
Conversely, on $(f_{k}\times f_{l})^{-1}(U)\subseteq X_{k}\times X_{l}$, the pullbacks of the universal families from $X_{k}$ and $X_{l}$ are disjoint and their union is a flat family of degree $n$. By the universal property there is an induced morphism $\psi \colon (f_{k}\times f_{l})^{-1}(U)\rightarrow H_{n}$, which factors through a morphism $\beta \colon (f_{k}\times f_{l})^{-1}(U)\rightarrow X_{n}$ of schemes over $(\mathbb{C}^{2})^{n}$.
By construction, the universal families on $f^{-1}(U)$ and $(f_{k}\times f_{l})^{-1}(U)$ pull back to themselves via $\beta \circ \alpha$ and $\alpha \circ \beta$, respectively. This implies that $\beta \circ \alpha$ is a morphism of schemes over $H_{n}$ and $\alpha \circ \beta$ is a morphism of schemes over $H_{k}\times H_{l}$. Since they are also morphisms of schemes over $(\mathbb{C}^{2})^{n}$, we have $\beta \circ \alpha = 1_{f^{-1}(U)}$ and $\alpha \circ \beta = 1_{(f_{k}\times f_{l})^{-1}(U)}$. Hence $\alpha$ and $\beta$ induce mutually inverse isomorphisms $f^{-1}(U)\cong (f_{k}\times f_{l})^{-1}(U)$.
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Proposition 3.3.2
The isospectral Hilbert scheme $X_{n}$ is irreducible, of dimension $2n$.
Proof.
Let $U$ be the preimage in $X_{n}$ of the open set $W\subseteq (\mathbb{C}^{2})^{n}$ consisting of points $(P_{1},\mathinner {\ldotp \ldotp \ldotp },P_{n})$ where the $P_{i}$ are all distinct. It follows from Lemma 3.3.1 that $f$ restricts to an isomorphism $f\colon U\rightarrow W$, so $U$ is irreducible. We are to show that $U$ is dense in $X_{n}$.
Let $Q$ be a closed point of $X_{n}$, which we want to show belongs to the closure $\overline{U}$ of $U$. If $f(Q) = (P_{1},\mathinner {\ldotp \ldotp \ldotp },P_{n})$ with the $P_{i}$ not all equal, then by Lemma 3.3.1 there is a neighborhood of $Q$ in $X_{n}$ isomorphic to an open set in $X_{k}\times X_{l}$ for some $k, l < n$. The result then follows by induction, since we may assume $X_{k}$ and $X_{l}$ irreducible. If all the $P_{i}$ are equal, then $Q$ is the unique point of $X_{n}$ lying over $I = \rho (Q)\in H_{n}$. Since $\rho$ is finite, $\rho (\overline{U})\subseteq H_{n}$ is closed. But $\rho (U)$ is dense in $H_{n}$, so $\rho (\overline{U}) = H_{n}$. Therefore $\overline{U}$ contains a point lying over $I$, which must be $Q$.
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Proposition 3.3.3
The closed subset $V(y_{1},\mathinner {\ldotp \ldotp \ldotp },y_{n})$ in $X_{n}$ has dimension $n$.
Proof.
It follows from the cell decomposition of Ellingsrud and Strømme Reference11 that the closed locus $Z$ in $H_{n}$ consisting of points $I$ with $V(I)$ supported on the $x$-axis$V(y)$ in $\mathbb{C}^{2}$ is the union of locally closed affine cells of dimension $n$. The subset $V(\mathbf{y})\subseteq X_{n}$ is equal to $\rho ^{-1}(Z)$ and $\rho$ is finite.
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The product structure of $X_{n}$ is inherited in a certain sense by $H_{n}$, but its description in terms of $X_{n}$ is more transparent. As a consequence, passage to $X_{n}$ is sometimes handy for proving results purely about $H_{n}$. The following lemma is an example of this. We remark that one can show by a more careful analysis that locus $G_{r}$ in the lemma is in fact irreducible.
Lemma 3.3.4
Let $G_{r}$ be the closed subset of $H_{n}$ consisting of ideals $I$ for which $\sigma (I)$ contains some point with multiplicity at least $r$. Then $G_{r}$ has codimension $r-1$, and has a unique irreducible component of maximal dimension.
Proof.
By symmetry among the points $P_{i}$ of $\sigma (I)$ we see that $G_{r} = \rho (V_{r})$, where $V_{r}$ is the locus in $X_{n}$ defined by the equations $P_{1}=\cdots =P_{r}$. It follows from Lemma 3.3.1 that $V_{r}\setminus \rho ^{-1}(G_{r+1})$ is isomorphic to an open set in $W_{r}\times X_{n-r}$, where $W_{r}$ is the closed subset $P_{1}=\cdots =P_{r}$ in $X_{r}$. As a reduced subscheme of $X_{r}$, the latter is isomorphic to $\mathbb{C}^{2}\times Z_{r}$, where $Z_{r} = \sigma ^{-1}(\underline {0})$ is the zero fiber in $H_{r}$, the factor $\mathbb{C}^{2}$ accounting for the choice of $P = P_{1}=\cdots =P_{r}$.
By a theorem of Briançon Reference5, $Z_{r}$ is irreducible of dimension $r-1$, so $V_{r}\setminus \rho ^{-1}(G_{r+1})$ is irreducible of dimension $2(n-r)+r+1 = 2n-(r-1)$. Since $G_{r}\setminus G_{r+1} = \rho (V_{r}\setminus \rho ^{-1}(G_{r+1}))$ and $\rho$ is finite, the result follows by descending induction on $r$, starting with $G_{n+1} = \emptyset$.
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3.4. Blowup construction of $H_{n}$ and $X_{n}$
Let $A = \mathbb{C}[\mathbf{x},\mathbf{y}]^{\epsilon }$ be the space of $S_{n}$-alternating elements, that is, polynomials $g$ such that $wg = \epsilon (w)g$ for all $w\in S_{n}$, where $\epsilon$ is the sign character. To describe $A$ more precisely, we note that $A$ is the image of the alternation operator
$$\begin{equation} \Theta ^{\epsilon }g = \sum _{w\in S_{n}} \epsilon (w) w g. \tag{21}\cssId{texmlid76}{} \end{equation}$$
If $D = \{ (p_{1},q_{1}),\mathinner {\ldotp \ldotp \ldotp },(p_{n},q_{n})\}$ is an $n$-element subset of $\mathbb{N}\times \mathbb{N}$, then the determinant $\Delta _{D}$ defined in Equation9 can also be written
where $\mathbf{x}^{p}\mathbf{y}^{q} = x_{1}^{p_{1}}y_{1}^{q_{1}}\cdots x_{n}^{p_{n}}y_{n}^{q_{n}}$. For a monomial $\mathbf{x}^{p} \mathbf{y}^{q}$ whose exponent pairs $(p_{i},q_{i})$ are not all distinct, we have $\Theta ^{\epsilon }(\mathbf{x}^{p}\mathbf{y}^{q}) = 0$. From this it is easy to see that the set of all elements $\Delta _{D}$ is a basis of $A$. Another way to see this is to identify $A$ with the $n$-th exterior power $\wedge ^{n}\mathbb{C}[x,y]$ of the polynomial ring in two variables $x,y$. Then the basis elements $\Delta _{D}$ are identified with the wedge products of monomials in $\mathbb{C}[x,y]$.
For $d>0$, let $A^{d}$ be the space spanned by all products of $d$ elements of $A$. We set $A^{0} = \mathbb{C}[\mathbf{x},\mathbf{y}]^{S_{n}}$. Note that $A$ and hence every $A^{d}$ is a $\mathbb{C}[\mathbf{x},\mathbf{y}]^{S_{n}}$-submodule of $\mathbb{C}[\mathbf{x},\mathbf{y}]$, so we have $A^{i}A^{j} = A^{i+j}$ for all $i$,$j$, including $i=0$ or $j=0$.
The Hilbert scheme $H_{n}$ is isomorphic as a scheme projective over $S^{n}\mathbb{C}^{2}$ to $\operatorname {Proj}T$, where $T$ is the graded $\mathbb{C}[\mathbf{x},\mathbf{y}]^{S_{n}}$-algebra$T = \bigoplus _{d\geq 0} A^{d}$.
Proposition 3.4.2
The isospectral Hilbert scheme $X_{n}$ is isomorphic as a scheme over $(\mathbb{C}^{2})^{n}$ to the blowup of $(\mathbb{C}^{2})^{n}$ at the ideal $J = \mathbb{C}[\mathbf{x},\mathbf{y}]A$ generated by the alternating polynomials.
Proof.
Set $S = \mathbb{C}[\mathbf{x},\mathbf{y}]$. By definition the blowup of $(\mathbb{C}^{2})^{n}$ at $J$ is $Z = \operatorname {Proj}S[tJ]$, where $S[tJ]\cong \bigoplus _{d\geq 0} J^{d}$ is the Rees algebra. The ring $T$ is a homogeneous subring of $S[tJ]$ in an obvious way, and since $A^{d}$ generates $J^{d}$ as a $\mathbb{C}[\mathbf{x},\mathbf{y}]$-module, we have $S\cdot T = S[tJ]$, that is, $S[tJ]\cong (\mathbb{C}[\mathbf{x},\mathbf{y}]\otimes _{A^{0}}T)/I$ for some homogeneous ideal $I$. In geometric terms, using Proposition 3.4.1 and the fact that $S^{n}\mathbb{C}^{2} = \operatorname {Spec}A^{0}$, this says that $Z$ is a closed subscheme of the scheme-theoretic fiber product $H_{n}\times _{S^{n}\mathbb{C}^{2}}(\mathbb{C}^{2})^{n}$. Since $Z$ is reduced, it follows that $Z$ is a closed subscheme of $X_{n}$. By Proposition 3.3.2, $X_{n}$ is irreducible, and since both $Z$ and $X_{n}$ have dimension $2n$, it follows that $Z=X_{n}$.
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In the context of either $H_{n}$ or $X_{n}$ we will always write $\mathcal{O}(k)$ for the $k$-th tensor power of the ample line bundle $\mathcal{O}(1)$ induced by the representation of $H_{n}$ as $\operatorname {Proj}T$ or $X_{n}$ as $\operatorname {Proj}S[tJ]$. It is immediate from the proof of Proposition 3.4.2 that $\mathcal{O}_{X_{n}}(k) = \rho ^{*}\mathcal{O}_{H_{n}}(k)$.
In full analogy to the situation for the Plücker embedding of a Grassmann variety, there is an intrinsic description of $\mathcal{O}(1)$ as the highest exterior power of the tautological vector bundle whose fiber at a point $I\in H_{n}$ is $R/I$. Let
be the projection of the universal family on the Hilbert scheme. Since $\pi$ is an affine morphism, we have $F = \operatorname {Spec}B$, where $B$ is the sheaf of $\mathcal{O}_{H_{n}}$-algebras
$$\begin{equation} B = \pi _{*}\mathcal{O}_{F}. \tag{24}\cssId{texmlid79}{} \end{equation}$$
The fact that $F$ is flat and finite of degree $n$ over $H_{n}$ means that $B$ is a locally free sheaf of $\mathcal{O}_{H_{n}}$-modules of rank $n$. Its associated vector bundle is the tautological bundle.
We have an isomorphism $\wedge ^{n} B \cong \mathcal{O}(1)$ of line bundles on $H_{n}$.
3.5. Nested Hilbert schemes
The proof of our main theorem will be by induction on $n$. For the inductive step we interpolate between $H_{n-1}$ and $H_{n}$ using the nested Hilbert scheme.
Definition 3.5.1
The nested Hilbert scheme$H_{n-1,n}$ 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 $H_{n-1,n}$ is nonsingular and irreducible, of dimension $2n$.
As with $H_{n}$, the nested Hilbert scheme is an open set in a projective nested Hilbert scheme $\operatorname {Hilb}^{n-1,n}(\mathbb{P}^{2})$. Clearly, $H_{n-1,n}$ is the preimage of $H_{n}$ under the projection $\operatorname {Hilb}^{n-1,n}(\mathbb{P}^{2})\rightarrow \operatorname {Hilb}^{n}(\mathbb{P}^{2})$. Hence the projection $H_{n-1,n}\rightarrow H_{n}$ is a projective morphism.
If $(I_{n-1},I_{n})$ is a point of $H_{n-1,n}$, then $\sigma (I_{n-1})$ is an $n-1$ element sub-multiset of $\sigma (I_{n})$. In symbols, if $\sigma (I_{n-1}) = \llbracket P_{1},\mathinner {\ldotp \ldotp \ldotp },P_{n-1} \rrbracket$, then $\sigma (I_{n}) = \llbracket P_{1},\mathinner {\ldotp \ldotp \ldotp },P_{n-1},P_{n} \rrbracket$ for a distinguished last point $P_{n}$. The $S_{n-1}$-invariant polynomials in the coordinates $x_{1},y_{1},\mathinner {\ldotp \ldotp \ldotp },x_{n-1},y_{n-1}$ of the points $P_{1},\mathinner {\ldotp \ldotp \ldotp },P_{n-1}$ are global regular functions on $H_{n-1,n}$, pulled back via the projection on $H_{n-1}$. Similarly, the $S_{n}$-invariant polynomials in $x_{1},y_{1},\mathinner {\ldotp \ldotp \ldotp },x_{n},y_{n}$ are regular functions pulled back from $H_{n}$. It follows that the coordinates of the distinguished point $P_{n}$ are regular functions, since $x_{n} = (x_{1}+\cdots +x_{n})-(x_{1}+\cdots +x_{n-1})$, and similarly for $y_{n}$. Thus we have a morphism
such that both the maps $H_{n-1,n}\rightarrow S^{n-1}\mathbb{C}^{2}$ and $H_{n-1,n}\rightarrow S^{n}\mathbb{C}^{2}$ induced by the Chow morphisms composed with the projections on $H_{n-1}$ and $H_{n}$ factor through $\sigma$.
The distinguished point $P_{n}$ belongs to $V(I_{n})$, and given $I_{n}$, every point of $V(I_{n})$ occurs as $P_{n}$ for some choice of $I_{n-1}$. Therefore the image of the morphism
sending $(I_{n-1},I_{n})$ to $(I_{n},P_{n})$ is precisely the universal family $F$ over $H_{n}$. For clarity, let us point out that by the definition of $F$, we have $F = \{(I,P)\in H_{n}\times \mathbb{C}^{2}:P\in V(I) \}$, at least set-theoretically. In fact, $F$ is reduced and hence coincides as a reduced closed subscheme with this subset of $H_{n}\times \mathbb{C}^{2}$. This is true because $F$ is flat over the variety $H_{n}$ 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 $H_{n-1,n}$ in §3.6 and the proof of the main theorem in §3.8.
Proposition 3.5.3
Let $d$ be the dimension of the fiber of the morphism $\alpha$ in Equation27 over a point $(I,P)\in F$, and let $r$ be the multiplicity of $P$ in $\sigma (I)$. Then $d$ and $r$ satisfy the inequality
Recall that the socle of an Artin local ring $A$ is the ideal consisting of elements annihilated by the maximal ideal $m$. If $A$ is an algebra over a field $k$, with $A/m\cong k$, then every linear subspace of the socle is an ideal, and conversely every ideal in $A$ of length $1$ is a one-dimensional subspace of the socle. The possible ideals $I_{n-1}$ for the given $(I_{n},P_{n}) = (I,P)$ are the length $1$ ideals in the Artin local $\mathbb{C}$-algebra$(R/I)_{P}$, where $R = \mathbb{C}[x,y]$. The fiber of $\alpha$ is therefore the projective space $\mathbb{P}(\operatorname {soc}(R/I)_{P})$, and we have ${d+1} = \dim \; \operatorname {soc}(R/I)_{P}$.
First consider the maximum possible dimension of any fiber of $\alpha$. Since both $H_{n-1,n}$ and $F$ are projective over $H_{n}$, the morphism $\alpha$ is projective and its fiber dimension is upper semicontinuous. Since every point of $H_{n}$ has a monomial ideal $I_{\mu }$ in the closure of its $\mathbb{T}^{2}$-orbit, and since $F$ is finite over $H_{n}$, every point of $F$ must have a pair $(I_{\mu },0)\in F$ in the closure of its orbit. The fiber dimension is therefore maximized at some such point. The socle of $R/I_{\mu }$ has dimension equal to the number of corners of the diagram of $\mu$. If this number is $s$, we clearly have $n\geq \binom {s+1}{2}$. This implies that for every Artin local $\mathbb{C}$-algebra$R/I$ generated by two elements, the socle dimension $s$ and the length $n$ of $R/I$ satisfy $n\geq \binom {s+1}{2}$.
Returning to the original problem, $(R/I)_{P}$ is an Artin local $\mathbb{C}$-algebra of length $r$ generated by two elements, with socle dimension $d+1$, so Equation28 follows.
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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 $n$: we transfer the Gorenstein property from $X_{n-1}$ to the nested scheme $X_{n-1,n}$ by pulling back, and from there to $X_{n}$ by pushing forward.
Definition 3.5.4
The nested isospectral Hilbert scheme$X_{n-1,n}$ is the reduced fiber product $H_{n-1,n}\times _{H_{n-1}} X_{n-1}$.
There is an alternative formulation of the definition, which is useful to keep in mind for the next two results. Namely, $X_{n-1,n}$ can be identified with the reduced fiber product in the diagram
that is, the reduced closed subscheme of $H_{n-1,n}\times (\mathbb{C}^{2})^{n}$ consisting of tuples $(I_{n-1}, I_{n}, P_{1},\mathinner {\ldotp \ldotp \ldotp }, P_{n})$ such that $\sigma (I_{n}) = \llbracket P_{1},\mathinner {\ldotp \ldotp \ldotp }, P_{n} \rrbracket$ and $P_{n}$ is the distinguished point. To see that this agrees with the definition, note that a point of $H_{n-1,n}\times _{H_{n-1}} X_{n-1}$ is given by the data $(I_{n-1},I_{n},P_{1},\mathinner {\ldotp \ldotp \ldotp },P_{n-1})$, and that these data determine the distinguished point $P_{n}$. We obtain the alternative description by identifying $X_{n-1,n}$ with the graph in $X_{n-1,n}\times \mathbb{C}^{2}$ of the morphism $X_{n-1,n}\rightarrow \mathbb{C}^{2}$ sending $(I_{n-1},I_{n},P_{1},\mathinner {\ldotp \ldotp \ldotp },P_{n-1})$ to $P_{n}$.
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., $X_{n-1,n}$ 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 $k+l = n$ and $U\subseteq (\mathbb{C}^{2})^{n}$ be as in Lemma 3.3.1. Then the preimage of $U$ in $X_{n-1,n}$ is isomorphic as a scheme over $(\mathbb{C}^{2})^{n}$ to the preimage of $U$ in $X_{k}\times X_{l-1,l}$.
Proof.
Lemma 3.3.1 gives us isomorphisms on the preimage of $U$ between $X_{n}$ and $X_{k}\times X_{l}$, and between $X_{n-1}$ and $X_{k}\times X_{l-1}$.
We can identify $X_{n-1,n}$ with the closed subset of $X_{n-1}\times X_{n}$ consisting of points where $P_{1},\mathinner {\ldotp \ldotp \ldotp },P_{n-1}$ are the same in both factors, and $I_{n-1}$ contains $I_{n}$. On the preimage of $U$, under the isomorphisms above, this corresponds to the closed subset of $(X_{k}\times X_{l-1})\times (X_{k}\times X_{l})$ where $P_{1},\mathinner {\ldotp \ldotp \ldotp },P_{k}$,$Q_{1},\mathinner {\ldotp \ldotp \ldotp },Q_{l-1}$ and $I_{k}$ are the same in both factors and $I_{l-1}$ contains $I_{l}$. The latter can be identified with $X_{k}\times X_{l-1,l}$.