American Mathematical Society

Hilbert schemes, polygraphs and the Macdonald positivity conjecture

By Mark Haiman

Abstract

We study the isospectral Hilbert scheme upper X Subscript n , defined as the reduced fiber product of left-parenthesis double-struck upper C squared right-parenthesis Superscript n with the Hilbert scheme upper H Subscript n of points in the plane double-struck upper C squared , over the symmetric power upper S Superscript n Baseline double-struck upper C squared equals left-parenthesis double-struck upper C squared right-parenthesis Superscript n Baseline slash upper S Subscript n . By a theorem of Fogarty, upper H Subscript n is smooth. We prove that upper X Subscript n is normal, Cohen-Macaulay and Gorenstein, and hence flat over upper H Subscript n . We derive two important consequences.

(1) We prove the strong form of the n factorial conjecture of Garsia and the author, giving a representation-theoretic interpretation of the Kostka-Macdonald coefficients upper K Subscript lamda mu Baseline left-parenthesis q comma t right-parenthesis . This establishes the Macdonald positivity conjecture, namely that upper K Subscript lamda mu Baseline left-parenthesis q comma t right-parenthesis element-of double-struck upper N left-bracket q comma t right-bracket .

(2) We show that the Hilbert scheme upper H Subscript n is isomorphic to the upper G -Hilbert scheme left-parenthesis double-struck upper C squared right-parenthesis Superscript n Baseline slash slash upper S Subscript n of Nakamura, in such a way that upper X Subscript n is identified with the universal family over left-parenthesis double-struck upper C squared right-parenthesis Superscript n Baseline slash slash upper S Subscript n . From this point of view, upper K Subscript lamda mu Baseline left-parenthesis q comma t right-parenthesis describes the fiber of a character sheaf upper C Subscript lamda at a torus-fixed point of left-parenthesis double-struck upper C squared right-parenthesis Superscript n Baseline slash slash upper S Subscript n corresponding to mu .

The proofs rely on a study of certain subspace arrangements upper Z left-parenthesis n comma l right-parenthesis subset-of-or-equal-to left-parenthesis double-struck upper C squared right-parenthesis Superscript n plus l , called polygraphs, whose coordinate rings upper R left-parenthesis n comma l right-parenthesis carry geometric information about upper X Subscript n . The key result is that upper R left-parenthesis n comma l right-parenthesis is a free module over the polynomial ring in one set of coordinates on left-parenthesis double-struck upper C squared right-parenthesis Superscript n . This is proven by an intricate inductive argument based on elementary commutative algebra.

1. Introduction

The Hilbert scheme of points in the plane upper H Subscript n Baseline equals upper H i l b Superscript n Baseline left-parenthesis double-struck upper C squared right-parenthesis is an algebraic variety which parametrizes finite subschemes upper S of length n in double-struck upper C squared . To each such subscheme upper S corresponds an n -element multiset, or unordered n -tuple with possible repetitions, sigma left-parenthesis upper S right-parenthesis equals mathematical left white bracket upper P 1 comma period period period comma upper P Subscript n Baseline mathematical right white bracket of points in double-struck upper C squared , where the upper P Subscript i are the points of upper S , repeated with appropriate multiplicities. There is a variety upper X Subscript n , finite over upper H Subscript n , whose fiber over the point of upper H Subscript n corresponding to upper S consists of all ordered n -tuples left-parenthesis upper P 1 comma period period period comma upper P Subscript n Baseline right-parenthesis element-of left-parenthesis double-struck upper C squared right-parenthesis Superscript n whose underlying multiset is sigma left-parenthesis upper S right-parenthesis . We call upper X Subscript n the isospectral Hilbert scheme.

By a theorem of Fogarty Reference14, the Hilbert scheme upper H Subscript n is irreducible and nonsingular. The geometry of upper X Subscript n is more complicated, but also very special. Our main geometric result, Theorem 3.1, is that upper X Subscript 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 upper H Subscript n and upper X Subscript n on the one hand, and the theory of Macdonald polynomials on the other. The Macdonald polynomials

StartLayout 1st Row with Label left-parenthesis 1 right-parenthesis EndLabel upper P Subscript mu Baseline left-parenthesis x semicolon q comma t right-parenthesis EndLayout

are a basis of the algebra of symmetric functions in variables x equals x 1 comma x 2 comma period period period , with coefficients in the field double-struck upper Q left-parenthesis q comma t right-parenthesis 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 upper K Subscript lamda mu Baseline left-parenthesis q comma t right-parenthesis , 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:

StartLayout 1st Row with Label left-parenthesis 2 right-parenthesis EndLabel upper K Subscript lamda mu Baseline left-parenthesis q comma t right-parenthesis element-of double-struck upper N left-bracket q comma t right-bracket period EndLayout

The positivity conjecture has remained open since Macdonald formulated it at the time of his original discovery. For q equals 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, upper K Subscript lamda mu Baseline left-parenthesis q comma t right-parenthesis element-of double-struck upper Z left-bracket q comma t right-bracket , but these results do not establish the positivity Reference18Reference19Reference32Reference33Reference44.

In Reference15, Garsia and the author conjectured an interpretation of the Kostka-Macdonald coefficients upper K Subscript lamda mu Baseline left-parenthesis q comma t right-parenthesis as graded character multiplicities for certain doubly graded upper S Subscript n -modules upper D Subscript mu . The module upper D Subscript mu is the space of polynomials in 2 n 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 upper K Subscript lamda mu Baseline left-parenthesis q comma t right-parenthesis , that for each partition mu of n , the dimension of upper D Subscript mu is equal to n factorial . This seemingly elementary assertion has come to be known as the n factorial 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 upper X Subscript n is equivalent to the n factorial conjecture. We further showed that the Cohen-Macaulay property of upper X Subscript n implies the stronger conjecture interpreting upper K Subscript lamda mu Baseline left-parenthesis q comma t right-parenthesis as a graded character multiplicity for upper D Subscript 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 upper H Subscript n is equal to the upper G -Hilbert scheme upper V double-solidus upper G of Ito and Nakamura Reference28, for the case upper V equals left-parenthesis double-struck upper C squared right-parenthesis Superscript n , upper G equals upper S Subscript n . The upper G -Hilbert scheme is of interest in connection with the generalized McKay correspondence, which says that if upper V is a complex vector space, upper G is a finite subgroup of upper S upper L left-parenthesis upper V right-parenthesis and upper Y right-arrow upper V slash upper G is a so-called crepant resolution of singularities, then the sum of the Betti numbers of upper Y equals the number of conjugacy classes of upper G . In many interesting cases Reference6Reference42, the upper G -Hilbert scheme turns out to be a crepant resolution and an instance of the McKay correspondence. By our main theorem, this holds for upper G equals upper S Subscript n , upper V equals left-parenthesis double-struck upper C squared right-parenthesis Superscript 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 upper K Subscript lamda mu Baseline left-parenthesis t right-parenthesis equals upper K Subscript lamda mu Baseline left-parenthesis 0 comma t right-parenthesis 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 upper K Subscript lamda mu Baseline left-parenthesis t right-parenthesis 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 factorial conjecture. Based on Garsia and Procesi’s experience, we initially expected that the n factorial conjecture itself would be easy to prove and that the difficulties would lie in the identification of upper K Subscript lamda mu Baseline left-parenthesis q comma t right-parenthesis as the graded character multiplicity. To our surprise, however, the n factorial conjecture stubbornly resisted elementary attack.

In the spring of 1992, we discussed our efforts on the n factorial conjecture with Procesi, along with another related conjecture we had stumbled upon in the course of our work. The modules involved in the n factorial conjecture are quotients of the ring upper R Subscript n of coinvariants for the action of upper S Subscript n on the polynomial ring in 2 n variables. This ring upper R Subscript n is isomorphic to the space of diagonal harmonics. Computations suggested that its dimension should be left-parenthesis n plus 1 right-parenthesis Superscript n minus 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 upper H Subscript n and what we now call the isospectral Hilbert scheme upper X Subscript n should be relevant to the determination of the dimension and character of upper R Subscript n . Specifically, he observed that there is a natural map from upper R Subscript n to the ring of global functions on the scheme-theoretic fiber in upper X Subscript n over the origin in upper S Superscript n Baseline double-struck upper C squared . With luck, this map might be an isomorphism, and—as we are now able to confirm— upper X Subscript n might be flat over upper H Subscript n , so that its structure sheaf would push down to a vector bundle on upper H Subscript n . Then upper R Subscript n would coincide with the space of global sections of this vector bundle over the zero-fiber in upper H Subscript n , and it might be possible to compute its character using the Atiyah-Bott Lefschetz formula.

The connection between upper X Subscript n and the n factorial 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 factorial conjecture should be the fibers of Procesi’s vector bundle at distinguished torus-fixed points in upper H Subscript n , a fact which we prove in §3.7. These considerations ultimately led to a conjectured formula for the character of upper R Subscript n in terms of Macdonald polynomials. This formula turned out to be correct up to the limit of practical computation ( n less-than-or-equal-to 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 upper X Subscript n should ultimately lead to a proof of the n factorial and Macdonald positivity conjectures, as is borne out here. By now the reader should expect the geometric study of upper X Subscript n also to yield a proof of the character formula for diagonal harmonics and the left-parenthesis n plus 1 right-parenthesis Superscript n minus 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 factorial 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 factorial 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 upper D Subscript 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 upper G -Hilbert schemes, along with related conjectures and open problems.

2. The n factorial and Macdonald positivity conjectures

2.1. Macdonald polynomials

We work with the transformed integral forms ModifyingAbove upper H With tilde Subscript mu Baseline left-parenthesis x semicolon q comma t right-parenthesis of the Macdonald polynomials, indexed by integer partitions mu , and homogeneous of degree n equals StartAbsoluteValue mu EndAbsoluteValue . These are defined as in Reference24, eq. (2.18), to be

StartLayout 1st Row with Label left-parenthesis 3 right-parenthesis EndLabel ModifyingAbove upper H With tilde Subscript mu Baseline left-parenthesis x semicolon q comma t right-parenthesis equals t Superscript n left-parenthesis mu right-parenthesis Baseline upper J Subscript mu Baseline left-bracket upper X slash left-parenthesis 1 minus t Superscript negative 1 Baseline right-parenthesis semicolon q comma t Superscript negative 1 Baseline right-bracket comma EndLayout

where upper J Subscript mu denotes Macdonald’s integral form as in Reference40, VI, eq. (8.3), and n left-parenthesis mu right-parenthesis is the partition statistic

StartLayout 1st Row with Label left-parenthesis 4 right-parenthesis EndLabel n left-parenthesis mu right-parenthesis equals sigma-summation Underscript i Endscripts left-parenthesis i minus 1 right-parenthesis mu Subscript i EndLayout

(not to be confused with n equals StartAbsoluteValue mu EndAbsoluteValue ).

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 double-struck upper F left-bracket left-bracket x right-bracket right-bracket be the algebra of formal series over the coefficient field double-struck upper F equals double-struck upper Q left-parenthesis q comma t right-parenthesis , in variables x equals x 1 comma x 2 comma period period period . For any upper A element-of double-struck upper F left-bracket left-bracket x right-bracket right-bracket , we denote by p Subscript k Baseline left-bracket upper A right-bracket the result of replacing each indeterminate in upper A by its k -th power. This includes the indeterminates q and t as well as the variables x Subscript i . The algebra of symmetric functions normal upper Lamda Subscript double-struck upper F is freely generated as an double-struck upper F -algebra by the power-sums

StartLayout 1st Row with Label left-parenthesis 5 right-parenthesis EndLabel p Subscript k Baseline left-parenthesis x right-parenthesis equals x 1 Superscript k Baseline plus x 2 Superscript k Baseline plus ellipsis period EndLayout

Hence there is a unique double-struck upper F -algebra homomorphism

StartLayout 1st Row with Label left-parenthesis 6 right-parenthesis EndLabel e v Subscript upper A Baseline colon normal upper Lamda Subscript double-struck upper F Baseline right-arrow double-struck upper F left-bracket left-bracket x right-bracket right-bracket defined by p Subscript k Baseline left-parenthesis x right-parenthesis right-arrow from bar p Subscript k Baseline left-bracket upper A right-bracket period EndLayout

In general we write f left-bracket upper A right-bracket for e v Subscript upper A Baseline left-parenthesis f right-parenthesis , for any f element-of normal upper Lamda Subscript double-struck upper F . With this notation goes the convention that upper X stands for the sum upper X equals x 1 plus x 2 plus ellipsis of the variables, so we have p Subscript k Baseline left-bracket upper X right-bracket equals p Subscript k Baseline left-parenthesis x right-parenthesis and hence f left-bracket upper X right-bracket equals f left-parenthesis x right-parenthesis for all f . Note that a plethystic substitution like f right-arrow from bar f left-bracket upper X slash left-parenthesis 1 minus t Superscript negative 1 Baseline right-parenthesis right-bracket , 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 upper H overTilde Subscript mu .

Proposition 2.1.1 (Reference24, Proposition 2.6).

The ModifyingAbove upper H With tilde Subscript mu Baseline left-parenthesis x semicolon q comma t right-parenthesis satisfy

(1)

ModifyingAbove upper H With tilde Subscript mu Baseline left-parenthesis x semicolon q comma t right-parenthesis element-of double-struck upper Q left-parenthesis q comma t right-parenthesis StartSet s Subscript lamda Baseline left-bracket upper X slash left-parenthesis 1 minus q right-parenthesis right-bracket colon lamda greater-than-or-equal-to mu EndSet ,

(2)

ModifyingAbove upper H With tilde Subscript mu Baseline left-parenthesis x semicolon q comma t right-parenthesis element-of double-struck upper Q left-parenthesis q comma t right-parenthesis StartSet s Subscript lamda Baseline left-bracket upper X slash left-parenthesis 1 minus t right-parenthesis right-bracket colon lamda greater-than-or-equal-to mu prime EndSet , and

(3)

ModifyingAbove upper H With tilde Subscript mu Baseline left-bracket 1 semicolon q comma t right-bracket equals 1 ,

where s Subscript lamda Baseline left-parenthesis x right-parenthesis denotes a Schur function, mu prime is the partition conjugate to mu , and the ordering is the dominance partial order on partitions of n equals StartAbsoluteValue mu EndAbsoluteValue . These conditions characterize ModifyingAbove upper H With tilde Subscript mu Baseline left-parenthesis x semicolon q comma t right-parenthesis uniquely.

We set ModifyingAbove upper K With tilde Subscript lamda mu Baseline left-parenthesis q comma t right-parenthesis equals t Superscript n left-parenthesis mu right-parenthesis Baseline upper K Subscript lamda mu Baseline left-parenthesis q comma t Superscript negative 1 Baseline right-parenthesis , where upper K Subscript lamda mu Baseline left-parenthesis q comma t right-parenthesis is the Kostka-Macdonald coefficient defined in Reference40, VI, eq. (8.11). This is then related to the transformed Macdonald polynomials by

StartLayout 1st Row with Label left-parenthesis 7 right-parenthesis EndLabel ModifyingAbove upper H With tilde Subscript mu Baseline left-parenthesis x semicolon q comma t right-parenthesis equals sigma-summation Underscript lamda Endscripts ModifyingAbove upper K With tilde Subscript lamda mu Baseline left-parenthesis q comma t right-parenthesis s Subscript lamda Baseline left-parenthesis x right-parenthesis period EndLayout

It is known that upper K Subscript lamda mu Baseline left-parenthesis q comma t right-parenthesis has degree at most n left-parenthesis mu right-parenthesis in t , so the positivity conjecture Equation2 from the introduction can be equivalently formulated in terms of upper K overTilde Subscript lamda mu .

Conjecture 2.1.2 (Macdonald positivity conjecture).

We have ModifyingAbove upper K With tilde Subscript lamda mu Baseline left-parenthesis q comma t right-parenthesis element-of double-struck upper N left-bracket q comma t right-bracket .

2.2. The n factorial and graded character conjectures

Let

StartLayout 1st Row with Label left-parenthesis 8 right-parenthesis EndLabel double-struck upper C left-bracket bold x comma bold y right-bracket equals double-struck upper C left-bracket x 1 comma y 1 comma period period period comma x Subscript n Baseline comma y Subscript n Baseline right-bracket EndLayout

be the polynomial ring in 2 n variables. To each n -element subset upper D subset-of-or-equal-to double-struck upper N times double-struck upper N , we associate a polynomial normal upper Delta Subscript upper D Baseline element-of double-struck upper C left-bracket bold x comma bold y right-bracket as follows. Let left-parenthesis p 1 comma q 1 right-parenthesis comma period period period comma left-parenthesis p Subscript n Baseline comma q Subscript n Baseline right-parenthesis be the elements of upper D listed in some fixed order. Then we define

StartLayout 1st Row with Label left-parenthesis 9 right-parenthesis EndLabel normal upper Delta Subscript upper D Baseline equals det left-parenthesis x Subscript i Superscript p Super Subscript j Superscript Baseline y Subscript i Superscript q Super Subscript j Superscript Baseline right-parenthesis Subscript 1 less-than-or-equal-to i comma j less-than-or-equal-to n Baseline period EndLayout

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

StartLayout 1st Row with Label left-parenthesis 10 right-parenthesis EndLabel upper D left-parenthesis mu right-parenthesis equals StartSet left-parenthesis i comma j right-parenthesis colon j less-than mu Subscript i plus 1 Baseline EndSet subset-of-or-equal-to double-struck upper N times double-struck upper N period EndLayout

(Note that in our definition the rows and columns of the diagram upper D left-parenthesis mu right-parenthesis are indexed starting with zero.) In the case where upper D equals upper D left-parenthesis mu right-parenthesis is the diagram of a partition, we abbreviate

StartLayout 1st Row with Label left-parenthesis 11 right-parenthesis EndLabel normal upper Delta Subscript mu Baseline equals normal upper Delta Subscript upper D left-parenthesis mu right-parenthesis Baseline period EndLayout

The polynomial normal upper Delta Subscript mu Baseline left-parenthesis bold x comma bold y right-parenthesis is a kind of bivariate analog of the Vandermonde determinant normal upper Delta left-parenthesis bold x right-parenthesis , which occurs as the special case mu equals left-parenthesis 1 Superscript n Baseline right-parenthesis .

Given a partition mu of n , we denote by

StartLayout 1st Row with Label left-parenthesis 12 right-parenthesis EndLabel upper D Subscript mu Baseline equals double-struck upper C left-bracket partial-differential bold x comma partial-differential bold y right-bracket normal upper Delta Subscript mu EndLayout

the space spanned by all the iterated partial derivatives of normal upper Delta Subscript 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 factorial conjecture).

The dimension of upper D Subscript mu is equal to n factorial .

The n factorial conjecture arose as part of a stronger conjecture relating the Kostka-Macdonald coefficients to the character of upper D Subscript mu as a doubly graded upper S Subscript n -module. The symmetric group upper S Subscript n acts by double-struck upper C -algebra automorphisms of double-struck upper C left-bracket bold x comma bold y right-bracket permuting the variables:

StartLayout 1st Row with Label left-parenthesis 13 right-parenthesis EndLabel w x Subscript i Baseline equals x Subscript w left-parenthesis i right-parenthesis Baseline comma w y Subscript i Baseline equals y Subscript w left-parenthesis i right-parenthesis Baseline for w element-of upper S Subscript n Baseline period EndLayout

The ring double-struck upper C left-bracket bold x comma bold y right-bracket equals circled-plus Underscript r comma s Endscripts double-struck upper C left-bracket bold x comma bold y right-bracket Subscript r comma s is doubly graded, by degree in the bold x and bold y variables respectively, and the upper S Subscript n action respects the grading. Clearly normal upper Delta Subscript mu is upper S Subscript n -alternating, i.e., we have w normal upper Delta Subscript mu Baseline equals epsilon left-parenthesis w right-parenthesis normal upper Delta Subscript mu for all w element-of upper S Subscript n , where epsilon is the sign character. Note that normal upper Delta Subscript mu is also doubly homogeneous, of x -degree n left-parenthesis mu right-parenthesis and y -degree n left-parenthesis mu prime right-parenthesis . It follows that the space upper D Subscript mu is upper S Subscript n -invariant and has a double grading

StartLayout 1st Row with Label left-parenthesis 14 right-parenthesis EndLabel upper D Subscript mu Baseline equals circled-plus Underscript r comma s Endscripts left-parenthesis upper D Subscript mu Baseline right-parenthesis Subscript r comma s EndLayout

by upper S Subscript n -invariant subspaces left-parenthesis upper D Subscript mu Baseline right-parenthesis Subscript r comma s Baseline equals upper D Subscript mu Baseline intersection double-struck upper C left-bracket bold x comma bold y right-bracket Subscript r comma s .

We write c h upper V for the character of an upper S Subscript n -module upper V , and denote the irreducible upper S Subscript n characters by chi Superscript lamda , with the usual indexing by partitions lamda of n . The following conjecture implies the Macdonald positivity conjecture.

Conjecture 2.2.2 (Reference15).

We have

StartLayout 1st Row with Label left-parenthesis 15 right-parenthesis EndLabel ModifyingAbove upper K With tilde Subscript lamda mu Baseline left-parenthesis q comma t right-parenthesis equals sigma-summation Underscript r comma s Endscripts t Superscript r Baseline q Superscript s Baseline mathematical left-angle chi Superscript lamda Baseline comma c h left-parenthesis upper D Subscript mu Baseline right-parenthesis Subscript r comma s Baseline mathematical right-angle period EndLayout

Macdonald had shown that upper K Subscript lamda mu Baseline left-parenthesis 1 comma 1 right-parenthesis is equal to chi Superscript lamda Baseline left-parenthesis 1 right-parenthesis , the degree of the irreducible upper S Subscript n character chi Superscript lamda , or the number of standard Young tableaux of shape lamda . Conjecture 2.2.2 therefore implies that upper D Subscript mu affords the regular representation of upper S Subscript n . In particular, it implies the n factorial conjecture.

In Reference24 the author showed that Conjecture 2.2.2 would follow from the Cohen-Macaulay property of upper X Subscript 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 upper X Subscript n , and deduce our main theorem, Theorem 3.13.8). We also define the Hilbert scheme upper H Subscript n and the nested Hilbert scheme upper H Subscript n minus 1 comma 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, double-struck upper C . All the specific schemes we consider are quasiprojective over double-struck upper 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 upper B of rank n on a scheme upper X of finite type over double-struck upper C is isomorphic to the sheaf of sections of an algebraic vector bundle of rank n over upper X . For notational purposes, we identify the vector bundle with the sheaf upper B and write upper B left-parenthesis x right-parenthesis for the fiber of upper B at a closed point x element-of upper X . In sheaf-theoretic terms, the fiber is given by upper B left-parenthesis x right-parenthesis equals upper B circled-times Subscript script upper O Sub Subscript upper X Baseline left-parenthesis script upper O Subscript upper X comma x Baseline slash x right-parenthesis .

A scheme upper X is Cohen-Macaulay or Gorenstein if its local ring script upper O Subscript upper X comma 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 upper H Subscript n and upper X Subscript n

Let upper R equals double-struck upper C left-bracket x comma y right-bracket be the coordinate ring of the affine plane double-struck upper C squared . By definition, closed subschemes upper S subset-of-or-equal-to double-struck upper C squared are in one-to-one correspondence with ideals upper I subset-of-or-equal-to upper R . The subscheme upper S equals upper V left-parenthesis upper I right-parenthesis is finite if and only if upper R slash upper I has Krull dimension zero, or finite dimension as a vector space over double-struck upper C . In this case, the length of upper S is defined to be dimension Subscript double-struck upper C Baseline upper R slash upper I .

The Hilbert scheme upper H Subscript n Baseline equals upper H i l b Superscript n Baseline left-parenthesis double-struck upper C squared right-parenthesis parametrizes finite closed subschemes upper S subset-of-or-equal-to double-struck upper C squared of length n . The scheme structure of upper H Subscript n and the precise sense in which it parametrizes the subschemes upper S are defined by a universal property, which characterizes upper H Subscript n up to unique isomorphism. The universal property is actually a property of upper H Subscript n together with a closed subscheme upper F subset-of-or-equal-to upper H Subscript n times double-struck upper C squared , called the universal family.

Proposition 3.2.1

There exist schemes upper H Subscript n Baseline equals upper H i l b Superscript n Baseline left-parenthesis double-struck upper C squared right-parenthesis and upper F subset-of-or-equal-to upper H Subscript n times double-struck upper C squared enjoying the following properties, which characterize them up to unique isomorphism:

(1)

upper F is flat and finite of degree n over upper H Subscript n , and

(2)

if upper Y subset-of-or-equal-to upper T times double-struck upper C squared is a closed subscheme, flat and finite of degree n over a scheme upper T , then there is a unique morphism phi colon upper T right-arrow upper H Subscript n Baseline giving a commutative fiber product diagram StartLayout 1st Row 1st Column upper Y 2nd Column right-arrow Overscript Endscripts 3rd Column Blank 4th Column upper T times double-struck upper C squared 5th Column Blank 6th Column right-arrow Overscript Endscripts 7th Column Blank 8th Column upper T 2nd Row 1st Column down-arrow 2nd Column Blank 3rd Column Blank 4th Column Blank 5th Column down-arrow 6th Column Blank 7th Column Blank 8th Column Blank 9th Column phi down-arrow 10th Column Blank 3rd Row 1st Column upper F 2nd Column right-arrow Overscript Endscripts 3rd Column Blank 4th Column upper H Subscript n Baseline times double-struck upper C squared 5th Column Blank 6th Column right-arrow Overscript Endscripts 7th Column Blank 8th Column upper H Subscript n Baseline comma EndLayout

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

Proof.

The Hilbert scheme ModifyingAbove upper H With caret equals upper H i l b Superscript n Baseline left-parenthesis double-struck upper P squared right-parenthesis of points in the projective plane exists as a special case of Grothendieck’s construction in Reference21, with a universal family ModifyingAbove upper F With caret having the analogous universal property. We identify double-struck upper C squared as usual with an open subset of double-struck upper P squared , the complement of the projective line upper Z “at infinity”.

The projection of ModifyingAbove upper F With caret intersection left-parenthesis ModifyingAbove upper H With caret times upper Z right-parenthesis onto ModifyingAbove upper H With caret is a closed subset of ModifyingAbove upper H With caret . Its complement upper H Subscript n Baseline subset-of-or-equal-to ModifyingAbove upper H With caret is clearly the largest subset such that the restriction upper F of ModifyingAbove upper F With caret to upper H Subscript n is contained in upper H Subscript n Baseline times double-struck upper C squared . The required universal property of upper H Subscript n and upper F now follows immediately from that of ModifyingAbove upper H With caret and ModifyingAbove upper F With caret .

To see how upper H Subscript n parametrizes finite closed subschemes upper S subset-of-or-equal-to double-struck upper C squared of length n , note that the latter are exactly the families upper Y in Proposition 3.2.1 for upper T equals upper S p e c double-struck upper C . By the universal property they correspond one-to-one with the closed points of upper H Subscript n , in such a way that the fiber of the universal family upper F over the point corresponding to upper S is upper S itself. For notational purposes we will identify the closed points of upper H Subscript n with ideals upper I subset-of-or-equal-to upper R satisfying dimension Subscript double-struck upper C Baseline upper R slash upper I equals n , rather than with the corresponding subschemes upper S equals upper V left-parenthesis upper I right-parenthesis .

We have the following fundamental theorem of Fogarty Reference14.

Proposition 3.2.2

The Hilbert scheme upper H Subscript n is a nonsingular, irreducible variety over double-struck upper C of dimension 2 n .

The generic examples of finite closed subschemes upper S subset-of-or-equal-to double-struck upper C squared of length n are the reduced subschemes consisting of n distinct points. They form an open subset of upper H Subscript 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 upper I subset-of-or-equal-to upper R is a monomial ideal, then the standard monomials x Superscript p Baseline y Superscript q not-an-element-of upper I form a basis of upper R slash upper I . If dimension Subscript double-struck upper C Baseline upper R slash upper I equals n , the exponents left-parenthesis p comma q right-parenthesis of the standard monomials form the diagram upper D left-parenthesis mu right-parenthesis of a partition mu of n , and conversely. We use the partition mu to index the corresponding monomial ideal, denoting it by upper I Subscript mu . Note that StartRoot EndRoot upper I Subscript mu Baseline equals left-parenthesis x comma y right-parenthesis for all mu , so the subscheme upper V left-parenthesis upper I Subscript mu Baseline right-parenthesis is concentrated at the origin left-parenthesis 0 comma 0 right-parenthesis element-of double-struck upper C squared , and owes its length entirely to its nonreduced scheme structure.

The algebraic torus

StartLayout 1st Row with Label left-parenthesis 16 right-parenthesis EndLabel double-struck upper T squared equals left-parenthesis double-struck upper C Superscript asterisk Baseline right-parenthesis squared EndLayout

acts on double-struck upper C squared as the group of invertible diagonal 2 times 2 matrices. The monomial ideals upper I Subscript mu are the torus invariant ideals, and thus they are the fixed points of the induced action of double-struck upper T squared on the Hilbert scheme. Every ideal upper I element-of upper H Subscript n has a monomial ideal in the closure of its double-struck upper T squared -orbit (Reference23, Lemma 2.3).

We write x Subscript i Baseline comma y Subscript i Baseline for the coordinates on the i -th factor in the Cartesian product left-parenthesis double-struck upper C squared right-parenthesis Superscript n , so we have left-parenthesis double-struck upper C squared right-parenthesis Superscript n Baseline equals upper S p e c double-struck upper C left-bracket bold x comma bold y right-bracket , where double-struck upper C left-bracket bold x comma bold y right-bracket equals double-struck upper C left-bracket x 1 comma y 1 comma period period period comma x Subscript n Baseline comma y Subscript n Baseline right-bracket . The symmetric group upper S Subscript n acts on left-parenthesis double-struck upper C squared right-parenthesis Superscript n by permuting the factors. In coordinates, this corresponds to the action of upper S Subscript n on double-struck upper C left-bracket bold x comma bold y right-bracket given in Equation13. We can identify upper S p e c double-struck upper C left-bracket bold x comma bold y right-bracket Superscript upper S Super Subscript n with the variety

StartLayout 1st Row with Label left-parenthesis 17 right-parenthesis EndLabel upper S Superscript n Baseline double-struck upper C squared equals left-parenthesis double-struck upper C squared right-parenthesis Superscript n Baseline slash upper S Subscript n EndLayout

of unordered n -tuples, or n -element multisets, of points in double-struck upper C squared .

Proposition 3.2.3 (Reference23, Proposition 2.2).

For upper I element-of upper H Subscript n , let sigma left-parenthesis upper I right-parenthesis be the multiset of points of upper V left-parenthesis upper I right-parenthesis , counting each point upper P with multiplicity equal to the length of the local ring left-parenthesis upper R slash upper I right-parenthesis Subscript upper P . Then the map sigma colon upper H Subscript n Baseline right-arrow upper S Superscript n Baseline double-struck upper C squared is a projective morphism (called the Chow morphism).

Definition 3.2.4

The isospectral Hilbert scheme upper X Subscript n is the reduced fiber product

StartLayout 1st Row with Label left-parenthesis 18 right-parenthesis EndLabel StartLayout 1st Row 1st Column upper X Subscript n 2nd Column right-arrow Overscript f Endscripts 3rd Column Blank 4th Column left-parenthesis double-struck upper C squared right-parenthesis Superscript n 2nd Row 1st Column rho down-arrow 2nd Column Blank 3rd Column Blank 4th Column Blank 5th Column down-arrow 6th Column Blank 3rd Row 1st Column upper H Subscript n 2nd Column right-arrow Overscript sigma Endscripts 3rd Column Blank 4th Column upper S Superscript n Baseline double-struck upper C squared comma EndLayout EndLayout

that is, the reduced closed subscheme of upper H Subscript n Baseline times left-parenthesis double-struck upper C squared right-parenthesis Superscript n whose closed points are the tuples left-parenthesis upper I comma upper P 1 comma period period period comma upper P Subscript n Baseline right-parenthesis satisfying sigma left-parenthesis upper I right-parenthesis equals mathematical left white bracket upper P 1 comma period period period comma upper P Subscript n Baseline mathematical right white bracket .

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

For each upper I element-of upper H Subscript n , the operators x overbar , y overbar of multiplication by x comma y are commuting endomorphisms of the n -dimensional vector space upper R slash upper I . As such, they have a well-defined joint spectrum, a multiset of pairs of eigenvalues left-parenthesis x 1 comma y 1 right-parenthesis comma period period period comma left-parenthesis x Subscript n Baseline comma y Subscript n Baseline right-parenthesis determined by the identity

StartLayout 1st Row with Label left-parenthesis 19 right-parenthesis EndLabel det Subscript upper R slash upper I Baseline left-parenthesis 1 plus alpha x overbar plus beta y overbar right-parenthesis equals product Underscript i equals 1 Overscript n Endscripts left-parenthesis 1 plus alpha x Subscript i Baseline plus beta y Subscript i Baseline right-parenthesis period EndLayout

On the local ring left-parenthesis upper R slash upper I right-parenthesis Subscript upper P at a point upper P equals left-parenthesis x 0 comma y 0 right-parenthesis , the operators x overbar , y overbar have the sole joint eigenvalue left-parenthesis x 0 comma y 0 right-parenthesis , with multiplicity equal to the length of left-parenthesis upper R slash upper I right-parenthesis Subscript upper P . Hence sigma left-parenthesis upper I right-parenthesis is equal as a multiset to the joint spectrum of x overbar and y overbar . This is the motivation for the term isospectral.

The action of upper S Subscript n on left-parenthesis double-struck upper C squared right-parenthesis Superscript n induces a compatible action of upper S Subscript n on upper X Subscript n by automorphisms of upper X Subscript n as a scheme over upper H Subscript n . Explicitly, for w element-of upper S Subscript n we have w left-parenthesis upper I comma upper P 1 comma period period period comma upper P Subscript n Baseline right-parenthesis equals left-parenthesis upper I comma upper P Subscript w Sub Superscript negative 1 Subscript left-parenthesis 1 right-parenthesis Baseline comma period period period comma upper P Subscript w Sub Superscript negative 1 Subscript left-parenthesis n right-parenthesis Baseline right-parenthesis .

We caution the reader that the scheme-theoretic fiber product in Equation18 is not reduced, even for n equals 2 . For every invariant polynomial g element-of double-struck upper C left-bracket bold x comma bold y right-bracket Superscript upper S Super Subscript n , the global regular function

StartLayout 1st Row with Label left-parenthesis 20 right-parenthesis EndLabel g left-parenthesis x 1 comma y 1 comma period period period comma x Subscript n Baseline comma y Subscript n Baseline right-parenthesis minus sigma Superscript asterisk Baseline g EndLayout

on upper H Subscript n Baseline times left-parenthesis double-struck upper C squared right-parenthesis Superscript n vanishes on upper X Subscript n . By definition these equations generate the ideal sheaf of the scheme-theoretic fiber product. They cut out upper X Subscript n set-theoretically, but not as a reduced subscheme. The full ideal sheaf defining upper X Subscript 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 upper J Subscript mu in §3.7, eq. Equation36, are implicit in the local ideals of upper X Subscript n at the distinguished points upper Q Subscript mu lying over the torus-fixed points upper I Subscript mu Baseline element-of upper H Subscript n .

3.3. Elementary properties of upper X Subscript n

We now develop some elementary facts about the isospectral Hilbert scheme upper X Subscript n . The first of these is its product structure, which allows us to reduce local questions on upper X Subscript n to questions about upper X Subscript k for k less-than n , in a neighborhood of any point whose corresponding multiset mathematical left white bracket upper P 1 comma period period period comma upper P Subscript n Baseline mathematical right white bracket is not of the form mathematical left white bracket n dot upper P mathematical right white bracket .

Lemma 3.3.1

Let k and l be positive integers with k plus l equals n . Suppose upper U subset-of-or-equal-to left-parenthesis double-struck upper C squared right-parenthesis Superscript n is an open set consisting of points left-parenthesis upper P 1 comma period period period comma upper P Subscript k Baseline comma upper Q 1 comma period period period comma upper Q Subscript l Baseline right-parenthesis where no upper P Subscript i coincides with any upper Q Subscript j . Then, identifying left-parenthesis double-struck upper C squared right-parenthesis Superscript n with left-parenthesis double-struck upper C squared right-parenthesis Superscript k Baseline times left-parenthesis double-struck upper C squared right-parenthesis Superscript l , the preimage f Superscript negative 1 Baseline left-parenthesis upper U right-parenthesis of upper U in upper X Subscript n is isomorphic as a scheme over left-parenthesis double-struck upper C squared right-parenthesis Superscript n to the preimage left-parenthesis f Subscript k Baseline times f Subscript l Baseline right-parenthesis Superscript negative 1 Baseline left-parenthesis upper U right-parenthesis of upper U in upper X Subscript k Baseline times upper X Subscript l .

Proof.

Let upper Y equals left-parenthesis rho times 1 Subscript double-struck upper C squared Baseline right-parenthesis Superscript negative 1 Baseline left-parenthesis upper F right-parenthesis subset-of-or-equal-to upper X Subscript n Baseline times double-struck upper C squared be the universal family over upper X Subscript n . The fiber upper V left-parenthesis upper I right-parenthesis of upper Y over a point left-parenthesis upper I comma upper P 1 comma period period period comma upper P Subscript k Baseline comma upper Q 1 comma period period period comma upper Q Subscript l Baseline right-parenthesis element-of f Superscript negative 1 Baseline left-parenthesis upper U right-parenthesis is the disjoint union of closed subschemes upper V left-parenthesis upper I Subscript k Baseline right-parenthesis and upper V left-parenthesis upper I Subscript l Baseline right-parenthesis in double-struck upper C squared of lengths k and l , respectively, with sigma left-parenthesis upper I Subscript k Baseline right-parenthesis equals mathematical left white bracket upper P 1 comma period period period comma upper P Subscript k Baseline mathematical right white bracket and sigma left-parenthesis upper I Subscript l Baseline right-parenthesis equals mathematical left white bracket upper Q 1 comma period period period comma upper Q Subscript l Baseline mathematical right white bracket . Hence over f Superscript negative 1 Baseline left-parenthesis upper U right-parenthesis , upper Y is the disjoint union of flat families upper Y Subscript k , upper Y Subscript l of degrees k and l . By the universal property, we get induced morphisms phi Subscript k Baseline colon f Superscript negative 1 Baseline left-parenthesis upper U right-parenthesis right-arrow upper H Subscript k Baseline , phi Subscript l Baseline colon f Superscript negative 1 Baseline left-parenthesis upper U right-parenthesis right-arrow upper H Subscript l Baseline and phi Subscript k Baseline times phi Subscript l Baseline colon f Superscript negative 1 Baseline left-parenthesis upper U right-parenthesis right-arrow upper H Subscript k Baseline times upper H Subscript l Baseline . The equations sigma left-parenthesis upper I Subscript k Baseline right-parenthesis equals mathematical left white bracket upper P 1 comma period period period comma upper P Subscript k Baseline mathematical right white bracket , sigma left-parenthesis upper I Subscript l Baseline right-parenthesis equals mathematical left white bracket upper Q 1 comma period period period comma upper Q Subscript l Baseline mathematical right white bracket imply that phi Subscript k Baseline times phi Subscript l factors through a morphism alpha colon f Superscript negative 1 Baseline left-parenthesis upper U right-parenthesis right-arrow upper X Subscript k Baseline times upper X Subscript l Baseline of schemes over left-parenthesis double-struck upper C squared right-parenthesis Superscript n .

Conversely, on left-parenthesis f Subscript k Baseline times f Subscript l Baseline right-parenthesis Superscript negative 1 Baseline left-parenthesis upper U right-parenthesis subset-of-or-equal-to upper X Subscript k times upper X Subscript l , the pullbacks of the universal families from upper X Subscript k and upper X Subscript l are disjoint and their union is a flat family of degree n . By the universal property there is an induced morphism psi colon left-parenthesis f Subscript k Baseline times f Subscript l Baseline right-parenthesis Superscript negative 1 Baseline left-parenthesis upper U right-parenthesis right-arrow upper H Subscript n Baseline , which factors through a morphism beta colon left-parenthesis f Subscript k Baseline times f Subscript l Baseline right-parenthesis Superscript negative 1 Baseline left-parenthesis upper U right-parenthesis right-arrow upper X Subscript n Baseline of schemes over left-parenthesis double-struck upper C squared right-parenthesis Superscript n .

By construction, the universal families on f Superscript negative 1 Baseline left-parenthesis upper U right-parenthesis and left-parenthesis f Subscript k Baseline times f Subscript l Baseline right-parenthesis Superscript negative 1 Baseline left-parenthesis upper U right-parenthesis pull back to themselves via beta ring alpha and alpha ring beta , respectively. This implies that beta ring alpha is a morphism of schemes over upper H Subscript n and alpha ring beta is a morphism of schemes over upper H Subscript k Baseline times upper H Subscript l . Since they are also morphisms of schemes over left-parenthesis double-struck upper C squared right-parenthesis Superscript n , we have beta ring alpha equals 1 Subscript f Sub Superscript negative 1 Subscript left-parenthesis upper U right-parenthesis and alpha ring beta equals 1 Subscript left-parenthesis f Sub Subscript k Subscript times f Sub Subscript l Subscript right-parenthesis Sub Superscript negative 1 Subscript left-parenthesis upper U right-parenthesis . Hence alpha and beta induce mutually inverse isomorphisms f Superscript negative 1 Baseline left-parenthesis upper U right-parenthesis approximately-equals left-parenthesis f Subscript k Baseline times f Subscript l Baseline right-parenthesis Superscript negative 1 Baseline left-parenthesis upper U right-parenthesis .

Proposition 3.3.2

The isospectral Hilbert scheme upper X Subscript n is irreducible, of dimension 2 n .

Proof.

Let upper U be the preimage in upper X Subscript n of the open set upper W subset-of-or-equal-to left-parenthesis double-struck upper C squared right-parenthesis Superscript n consisting of points left-parenthesis upper P 1 comma period period period comma upper P Subscript n Baseline right-parenthesis where the upper P Subscript i are all distinct. It follows from Lemma 3.3.1 that f restricts to an isomorphism f colon upper U right-arrow upper W , so upper U is irreducible. We are to show that upper U is dense in upper X Subscript n .

Let upper Q be a closed point of upper X Subscript n , which we want to show belongs to the closure upper U overbar of upper U . If f left-parenthesis upper Q right-parenthesis equals left-parenthesis upper P 1 comma period period period comma upper P Subscript n Baseline right-parenthesis with the upper P Subscript i not all equal, then by Lemma 3.3.1 there is a neighborhood of upper Q in upper X Subscript n isomorphic to an open set in upper X Subscript k Baseline times upper X Subscript l for some k comma l less-than n . The result then follows by induction, since we may assume upper X Subscript k and upper X Subscript l irreducible. If all the upper P Subscript i are equal, then upper Q is the unique point of upper X Subscript n lying over upper I equals rho left-parenthesis upper Q right-parenthesis element-of upper H Subscript n . Since rho is finite, rho left-parenthesis upper U overbar right-parenthesis subset-of-or-equal-to upper H Subscript n is closed. But rho left-parenthesis upper U right-parenthesis is dense in upper H Subscript n , so rho left-parenthesis upper U overbar right-parenthesis equals upper H Subscript n . Therefore upper U overbar contains a point lying over upper I , which must be upper Q .

Proposition 3.3.3

The closed subset upper V left-parenthesis y 1 comma period period period comma y Subscript n Baseline right-parenthesis in upper X Subscript n has dimension n .

Proof.

It follows from the cell decomposition of Ellingsrud and Strømme Reference11 that the closed locus upper Z in upper H Subscript n consisting of points upper I with upper V left-parenthesis upper I right-parenthesis supported on the x -axis upper V left-parenthesis y right-parenthesis in double-struck upper C squared is the union of locally closed affine cells of dimension n . The subset upper V left-parenthesis bold y right-parenthesis subset-of-or-equal-to upper X Subscript n is equal to rho Superscript negative 1 Baseline left-parenthesis upper Z right-parenthesis and rho is finite.

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

Lemma 3.3.4

Let upper G Subscript r be the closed subset of upper H Subscript n consisting of ideals upper I for which sigma left-parenthesis upper I right-parenthesis contains some point with multiplicity at least r . Then upper G Subscript r has codimension r minus 1 , and has a unique irreducible component of maximal dimension.

Proof.

By symmetry among the points upper P Subscript i of sigma left-parenthesis upper I right-parenthesis we see that upper G Subscript r Baseline equals rho left-parenthesis upper V Subscript r Baseline right-parenthesis , where upper V Subscript r is the locus in upper X Subscript n defined by the equations upper P 1 equals ellipsis equals upper P Subscript r . It follows from Lemma 3.3.1 that upper V Subscript r Baseline minus rho Superscript negative 1 Baseline left-parenthesis upper G Subscript r plus 1 Baseline right-parenthesis is isomorphic to an open set in upper W Subscript r Baseline times upper X Subscript n minus r , where upper W Subscript r is the closed subset upper P 1 equals ellipsis equals upper P Subscript r in upper X Subscript r . As a reduced subscheme of upper X Subscript r , the latter is isomorphic to double-struck upper C squared times upper Z Subscript r , where upper Z Subscript r Baseline equals sigma Superscript negative 1 Baseline left-parenthesis ModifyingBelow 0 With bar right-parenthesis is the zero fiber in upper H Subscript r , the factor double-struck upper C squared accounting for the choice of upper P equals upper P 1 equals ellipsis equals upper P Subscript r .

By a theorem of Briançon Reference5, upper Z Subscript r is irreducible of dimension r minus 1 , so upper V Subscript r Baseline minus rho Superscript negative 1 Baseline left-parenthesis upper G Subscript r plus 1 Baseline right-parenthesis is irreducible of dimension 2 left-parenthesis n minus r right-parenthesis plus r plus 1 equals 2 n minus left-parenthesis r minus 1 right-parenthesis . Since upper G Subscript r Baseline minus upper G Subscript r plus 1 Baseline equals rho left-parenthesis upper V Subscript r Baseline minus rho Superscript negative 1 Baseline left-parenthesis upper G Subscript r plus 1 Baseline right-parenthesis right-parenthesis and rho is finite, the result follows by descending induction on r , starting with upper G Subscript n plus 1 Baseline equals normal empty-set .

3.4. Blowup construction of upper H Subscript n and upper X Subscript n

Let upper A equals double-struck upper C left-bracket bold x comma bold y right-bracket Superscript epsilon be the space of upper S Subscript n -alternating elements, that is, polynomials g such that w g equals epsilon left-parenthesis w right-parenthesis g for all w element-of upper S Subscript n , where epsilon is the sign character. To describe upper A more precisely, we note that upper A is the image of the alternation operator

StartLayout 1st Row with Label left-parenthesis 21 right-parenthesis EndLabel normal upper Theta Superscript epsilon Baseline g equals sigma-summation Underscript w element-of upper S Subscript n Baseline Endscripts epsilon left-parenthesis w right-parenthesis w g period EndLayout

If upper D equals StartSet left-parenthesis p 1 comma q 1 right-parenthesis comma period period period comma left-parenthesis p Subscript n Baseline comma q Subscript n Baseline right-parenthesis EndSet is an n -element subset of double-struck upper N times double-struck upper N , then the determinant normal upper Delta Subscript upper D defined in Equation9 can also be written

StartLayout 1st Row with Label left-parenthesis 22 right-parenthesis EndLabel normal upper Delta Subscript upper D Baseline equals normal upper Theta Superscript epsilon Baseline left-parenthesis bold x Superscript p Baseline bold y Superscript q Baseline right-parenthesis comma EndLayout

where bold x Superscript p Baseline bold y Superscript q Baseline equals x 1 Superscript p 1 Baseline y 1 Superscript q 1 Baseline ellipsis x Subscript n Superscript p Super Subscript n Baseline y Subscript n Superscript q Super Subscript n . For a monomial bold x Superscript p Baseline bold y Superscript q whose exponent pairs left-parenthesis p Subscript i Baseline comma q Subscript i Baseline right-parenthesis are not all distinct, we have normal upper Theta Superscript epsilon Baseline left-parenthesis bold x Superscript p Baseline bold y Superscript q Baseline right-parenthesis equals 0 . From this it is easy to see that the set of all elements normal upper Delta Subscript upper D is a basis of upper A . Another way to see this is to identify upper A with the n -th exterior power logical-and double-struck upper C left-bracket x comma y right-bracket of the polynomial ring in two variables x comma y . Then the basis elements normal upper Delta Subscript upper D are identified with the wedge products of monomials in double-struck upper C left-bracket x comma y right-bracket .

For d greater-than 0 , let upper A Superscript d be the space spanned by all products of d elements of upper A . We set upper A Superscript 0 Baseline equals double-struck upper C left-bracket bold x comma bold y right-bracket Superscript upper S Super Subscript n . Note that upper A and hence every upper A Superscript d is a double-struck upper C left-bracket bold x comma bold y right-bracket Superscript upper S Super Subscript n -submodule of double-struck upper C left-bracket bold x comma bold y right-bracket , so we have upper A Superscript i Baseline upper A Superscript j Baseline equals upper A Superscript i plus j for all i , j , including i equals 0 or j equals 0 .

Proposition 3.4.1 (Reference23, Proposition 2.6).

The Hilbert scheme upper H Subscript n is isomorphic as a scheme projective over upper S Superscript n Baseline double-struck upper C squared to upper P r o j upper T , where upper T is the graded double-struck upper C left-bracket bold x comma bold y right-bracket Superscript upper S Super Subscript n -algebra upper T equals circled-plus Underscript d greater-than-or-equal-to 0 Endscripts upper A Superscript d .

Proposition 3.4.2

The isospectral Hilbert scheme upper X Subscript n is isomorphic as a scheme over left-parenthesis double-struck upper C squared right-parenthesis Superscript n to the blowup of left-parenthesis double-struck upper C squared right-parenthesis Superscript n at the ideal upper J equals double-struck upper C left-bracket bold x comma bold y right-bracket upper A generated by the alternating polynomials.

Proof.

Set upper S equals double-struck upper C left-bracket bold x comma bold y right-bracket . By definition the blowup of left-parenthesis double-struck upper C squared right-parenthesis Superscript n at upper J is upper Z equals upper P r o j upper S left-bracket t upper J right-bracket , where upper S left-bracket t upper J right-bracket approximately-equals circled-plus Underscript d greater-than-or-equal-to 0 Endscripts upper J Superscript d is the Rees algebra. The ring upper T is a homogeneous subring of upper S left-bracket t upper J right-bracket in an obvious way, and since upper A Superscript d generates upper J Superscript d as a double-struck upper C left-bracket bold x comma bold y right-bracket -module, we have upper S dot upper T equals upper S left-bracket t upper J right-bracket , that is, upper S left-bracket t upper J right-bracket approximately-equals left-parenthesis double-struck upper C left-bracket bold x comma bold y right-bracket circled-times Subscript upper A Sub Superscript 0 Subscript Baseline upper T right-parenthesis slash upper I for some homogeneous ideal upper I . In geometric terms, using Proposition 3.4.1 and the fact that upper S Superscript n Baseline double-struck upper C squared equals upper S p e c upper A Superscript 0 , this says that upper Z is a closed subscheme of the scheme-theoretic fiber product upper H Subscript n Baseline times Subscript upper S Sub Superscript n Subscript double-struck upper C squared Baseline left-parenthesis double-struck upper C squared right-parenthesis Superscript n . Since upper Z is reduced, it follows that upper Z is a closed subscheme of upper X Subscript n . By Proposition 3.3.2, upper X Subscript n is irreducible, and since both upper Z and upper X Subscript n have dimension 2 n , it follows that upper Z equals upper X Subscript n .

In the context of either upper H Subscript n or upper X Subscript n we will always write script upper O left-parenthesis k right-parenthesis for the k -th tensor power of the ample line bundle script upper O left-parenthesis 1 right-parenthesis induced by the representation of upper H Subscript n as upper P r o j upper T or upper X Subscript n as upper P r o j upper S left-bracket t upper J right-bracket . It is immediate from the proof of Proposition 3.4.2 that script upper O Subscript upper X Sub Subscript n Baseline left-parenthesis k right-parenthesis equals rho Superscript asterisk Baseline script upper O Subscript upper H Sub Subscript n Baseline left-parenthesis k right-parenthesis .

In full analogy to the situation for the Plücker embedding of a Grassmann variety, there is an intrinsic description of script upper O left-parenthesis 1 right-parenthesis as the highest exterior power of the tautological vector bundle whose fiber at a point upper I element-of upper H Subscript n is upper R slash upper I . Let

StartLayout 1st Row with Label left-parenthesis 23 right-parenthesis EndLabel pi colon upper F right-arrow upper H Subscript n Baseline EndLayout

be the projection of the universal family on the Hilbert scheme. Since pi is an affine morphism, we have upper F equals upper S p e c upper B , where upper B is the sheaf of script upper O Subscript upper H Sub Subscript n -algebras

StartLayout 1st Row with Label left-parenthesis 24 right-parenthesis EndLabel upper B equals pi Subscript asterisk Baseline script upper O Subscript upper F Baseline period EndLayout

The fact that upper F is flat and finite of degree n over upper H Subscript n means that upper B is a locally free sheaf of script upper O Subscript upper H Sub Subscript n -modules of rank n . Its associated vector bundle is the tautological bundle.

Proposition 3.4.3 (Reference23, Proposition 2.12).

We have an isomorphism logical-and upper B approximately-equals script upper O left-parenthesis 1 right-parenthesis of line bundles on upper H Subscript 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 upper H Subscript n minus 1 and upper H Subscript n using the nested Hilbert scheme.

Definition 3.5.1

The nested Hilbert scheme upper H Subscript n minus 1 comma n is the reduced closed subscheme

StartLayout 1st Row with Label left-parenthesis 25 right-parenthesis EndLabel upper H Subscript n minus 1 comma n Baseline equals StartSet left-parenthesis upper I Subscript n minus 1 Baseline comma upper I Subscript n Baseline right-parenthesis colon upper I Subscript n Baseline subset-of-or-equal-to upper I Subscript n minus 1 Baseline EndSet subset-of-or-equal-to upper H Subscript n minus 1 Baseline times upper H Subscript n Baseline period EndLayout

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 upper H Subscript n minus 1 comma n is nonsingular and irreducible, of dimension 2 n .

As with upper H Subscript n , the nested Hilbert scheme is an open set in a projective nested Hilbert scheme upper H i l b Superscript n minus 1 comma n Baseline left-parenthesis double-struck upper P squared right-parenthesis . Clearly, upper H Subscript n minus 1 comma n is the preimage of upper H Subscript n under the projection upper H i l b Superscript n minus 1 comma n Baseline left-parenthesis double-struck upper P squared right-parenthesis right-arrow upper H i l b Superscript n Baseline left-parenthesis double-struck upper P squared right-parenthesis . Hence the projection upper H Subscript n minus 1 comma n Baseline right-arrow upper H Subscript n is a projective morphism.

If left-parenthesis upper I Subscript n minus 1 Baseline comma upper I Subscript n Baseline right-parenthesis is a point of upper H Subscript n minus 1 comma n , then sigma left-parenthesis upper I Subscript n minus 1 Baseline right-parenthesis is an n minus 1 element sub-multiset of sigma left-parenthesis upper I Subscript n Baseline right-parenthesis . In symbols, if sigma left-parenthesis upper I Subscript n minus 1 Baseline right-parenthesis equals mathematical left white bracket upper P 1 comma period period period comma upper P Subscript n minus 1 Baseline mathematical right white bracket , then sigma left-parenthesis upper I Subscript n Baseline right-parenthesis equals mathematical left white bracket upper P 1 comma period period period comma upper P Subscript n minus 1 Baseline comma upper P Subscript n Baseline mathematical right white bracket for a distinguished last point upper P Subscript n . The upper S Subscript n minus 1 -invariant polynomials in the coordinates x 1 comma y 1 comma period period period comma x Subscript n minus 1 Baseline comma y Subscript n minus 1 Baseline of the points upper P 1 comma period period period comma upper P Subscript n minus 1 Baseline are global regular functions on upper H Subscript n minus 1 comma n , pulled back via the projection on upper H Subscript n minus 1 . Similarly, the upper S Subscript n -invariant polynomials in x 1 comma y 1 comma period period period comma x Subscript n Baseline comma y Subscript n Baseline are regular functions pulled back from upper H Subscript n . It follows that the coordinates of the distinguished point upper P Subscript n are regular functions, since x Subscript n Baseline equals left-parenthesis x 1 plus ellipsis plus x Subscript n Baseline right-parenthesis minus left-parenthesis x 1 plus ellipsis plus x Subscript n minus 1 Baseline right-parenthesis , and similarly for y Subscript n . Thus we have a morphism

StartLayout 1st Row with Label left-parenthesis 26 right-parenthesis EndLabel sigma colon upper H Subscript n minus 1 comma n Baseline right-arrow upper S Superscript n minus 1 Baseline double-struck upper C squared times double-struck upper C squared equals left-parenthesis double-struck upper C squared right-parenthesis Superscript n Baseline slash upper S Subscript n minus 1 Baseline comma EndLayout

such that both the maps upper H Subscript n minus 1 comma n Baseline right-arrow upper S Superscript n minus 1 Baseline double-struck upper C squared and upper H Subscript n minus 1 comma n Baseline right-arrow upper S Superscript n Baseline double-struck upper C squared induced by the Chow morphisms composed with the projections on upper H Subscript n minus 1 and upper H Subscript n factor through sigma .

The distinguished point upper P Subscript n belongs to upper V left-parenthesis upper I Subscript n Baseline right-parenthesis , and given upper I Subscript n , every point of upper V left-parenthesis upper I Subscript n Baseline right-parenthesis occurs as upper P Subscript n for some choice of upper I Subscript n minus 1 . Therefore the image of the morphism

StartLayout 1st Row with Label left-parenthesis 27 right-parenthesis EndLabel alpha colon upper H Subscript n minus 1 comma n Baseline right-arrow upper H Subscript n Baseline times double-struck upper C squared EndLayout

sending left-parenthesis upper I Subscript n minus 1 Baseline comma upper I Subscript n Baseline right-parenthesis to left-parenthesis upper I Subscript n Baseline comma upper P Subscript n Baseline right-parenthesis is precisely the universal family upper F over upper H Subscript n . For clarity, let us point out that by the definition of upper F , we have upper F equals StartSet left-parenthesis upper I comma upper P right-parenthesis element-of upper H Subscript n Baseline times double-struck upper C squared colon upper P element-of upper V left-parenthesis upper I right-parenthesis EndSet , at least set-theoretically. In fact, upper F is reduced and hence coincides as a reduced closed subscheme with this subset of upper H Subscript n Baseline times double-struck upper C squared . This is true because upper F is flat over the variety upper H Subscript 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 upper H Subscript n minus 1 comma 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 left-parenthesis upper I comma upper P right-parenthesis element-of upper F , and let r be the multiplicity of upper P in sigma left-parenthesis upper I right-parenthesis . Then d and r satisfy the inequality

StartLayout 1st Row with Label left-parenthesis 28 right-parenthesis EndLabel r greater-than-or-equal-to StartBinomialOrMatrix d plus 2 Choose 2 EndBinomialOrMatrix period EndLayout

Proof.

Recall that the socle of an Artin local ring upper A is the ideal consisting of elements annihilated by the maximal ideal m . If upper A is an algebra over a field k , with upper A slash m approximately-equals k , then every linear subspace of the socle is an ideal, and conversely every ideal in upper A of length 1 is a one-dimensional subspace of the socle. The possible ideals upper I Subscript n minus 1 for the given left-parenthesis upper I Subscript n Baseline comma upper P Subscript n Baseline right-parenthesis equals left-parenthesis upper I comma upper P right-parenthesis are the length 1 ideals in the Artin local double-struck upper C -algebra left-parenthesis upper R slash upper I right-parenthesis Subscript upper P , where upper R equals double-struck upper C left-bracket x comma y right-bracket . The fiber of alpha is therefore the projective space double-struck upper P left-parenthesis s o c left-parenthesis upper R slash upper I right-parenthesis Subscript upper P Baseline right-parenthesis , and we have d plus 1 equals dimension s o c left-parenthesis upper R slash upper I right-parenthesis Subscript upper P .

First consider the maximum possible dimension of any fiber of alpha . Since both upper H Subscript n minus 1 comma n and upper F are projective over upper H Subscript n , the morphism alpha is projective and its fiber dimension is upper semicontinuous. Since every point of upper H Subscript n has a monomial ideal upper I Subscript mu in the closure of its double-struck upper T squared -orbit, and since upper F is finite over upper H Subscript n , every point of upper F must have a pair left-parenthesis upper I Subscript mu Baseline comma 0 right-parenthesis element-of upper F in the closure of its orbit. The fiber dimension is therefore maximized at some such point. The socle of upper R slash upper I Subscript mu has dimension equal to the number of corners of the diagram of mu . If this number is s , we clearly have n greater-than-or-equal-to StartBinomialOrMatrix s plus 1 Choose 2 EndBinomialOrMatrix . This implies that for every Artin local double-struck upper C -algebra upper R slash upper I generated by two elements, the socle dimension s and the length n of upper R slash upper I satisfy n greater-than-or-equal-to StartBinomialOrMatrix s plus 1 Choose 2 EndBinomialOrMatrix .

Returning to the original problem, left-parenthesis upper R slash upper I right-parenthesis Subscript upper P is an Artin local double-struck upper C -algebra of length r generated by two elements, with socle dimension d plus 1 , 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 n : we transfer the Gorenstein property from upper X Subscript n minus 1 to the nested scheme upper X Subscript n minus 1 comma n by pulling back, and from there to upper X Subscript n by pushing forward.

Definition 3.5.4

The nested isospectral Hilbert scheme upper X Subscript n minus 1 comma n is the reduced fiber product upper H Subscript n minus 1 comma n Baseline times Subscript upper H Sub Subscript n minus 1 Baseline upper X Subscript n minus 1 .

There is an alternative formulation of the definition, which is useful to keep in mind for the next two results. Namely, upper X Subscript n minus 1 comma n can be identified with the reduced fiber product in the diagram

StartLayout 1st Row with Label left-parenthesis 29 right-parenthesis EndLabel StartLayout 1st Row 1st Column upper X Subscript n minus 1 comma n 2nd Column right-arrow Overscript Endscripts 3rd Column Blank 4th Column left-parenthesis double-struck upper C squared right-parenthesis Superscript n 2nd Row 1st Column down-arrow 2nd Column Blank 3rd Column Blank 4th Column Blank 5th Column down-arrow 6th Column Blank 3rd Row 1st Column upper H Subscript n minus 1 comma n 2nd Column right-arrow Overscript sigma Endscripts 3rd Column Blank 4th Column upper S Superscript n minus 1 Baseline double-struck upper C squared times double-struck upper C squared comma EndLayout EndLayout

that is, the reduced closed subscheme of upper H Subscript n minus 1 comma n Baseline times left-parenthesis double-struck upper C squared right-parenthesis Superscript n consisting of tuples left-parenthesis upper I Subscript n minus 1 Baseline comma upper I Subscript n Baseline comma upper P 1 comma period period period comma upper P Subscript n Baseline right-parenthesis such that sigma left-parenthesis upper I Subscript n Baseline right-parenthesis equals mathematical left white bracket upper P 1 comma period period period comma upper P Subscript n Baseline mathematical right white bracket and upper P Subscript n is the distinguished point. To see that this agrees with the definition, note that a point of upper H Subscript n minus 1 comma n Baseline times Subscript upper H Sub Subscript n minus 1 Baseline upper X Subscript n minus 1 is given by the data left-parenthesis upper I Subscript n minus 1 Baseline comma upper I Subscript n Baseline comma upper P 1 comma period period period comma upper P Subscript n minus 1 Baseline right-parenthesis , and that these data determine the distinguished point upper P Subscript n . We obtain the alternative description by identifying upper X Subscript n minus 1 comma n with the graph in upper X Subscript n minus 1 comma n Baseline times double-struck upper C squared of the morphism upper X Subscript n minus 1 comma n Baseline right-arrow double-struck upper C squared sending left-parenthesis upper I Subscript n minus 1 Baseline comma upper I Subscript n Baseline comma upper P 1 comma period period period comma upper P Subscript n minus 1 Baseline right-parenthesis to upper P Subscript 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., upper X Subscript n minus 1 comma 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 plus l equals n and upper U subset-of-or-equal-to left-parenthesis double-struck upper C squared right-parenthesis Superscript n be as in Lemma 3.3.1. Then the preimage of upper U in upper X Subscript n minus 1 comma n is isomorphic as a scheme over left-parenthesis double-struck upper C squared right-parenthesis Superscript n to the preimage of upper U in upper X Subscript k Baseline times upper X Subscript l minus 1 comma l .

Proof.

Lemma 3.3.1 gives us isomorphisms on the preimage of upper U between upper X Subscript n and upper X Subscript k Baseline times upper X Subscript l , and between upper X Subscript n minus 1 and upper X Subscript k Baseline times upper X Subscript l minus 1 .

We can identify upper X Subscript n minus 1 comma n with the closed subset of upper X Subscript n minus 1 Baseline times upper X Subscript n consisting of points where upper P 1 comma period period period comma upper P Subscript n minus 1 Baseline are the same in both factors, and upper I Subscript n minus 1 contains upper I Subscript n . On the preimage of upper U , under the isomorphisms above, this corresponds to the closed subset of left-parenthesis upper X Subscript k Baseline times upper X Subscript l minus 1 Baseline right-parenthesis times left-parenthesis upper X Subscript k Baseline times upper X Subscript l Baseline right-parenthesis where upper P 1 comma period period period comma upper P Subscript k Baseline , upper Q 1 comma period period period comma upper Q Subscript l minus 1 Baseline and upper I Subscript k are the same in both factors and upper I Subscript l minus 1 contains upper I Subscript l . The latter can be identified with upper X Subscript k Baseline times upper X Subscript l minus 1 comma l .

Proposition 3.5.6

The closed subset upper V left-parenthesis y 1 comma period period period comma y Subscript n Baseline right-parenthesis in upper X Subscript n minus 1 comma n has dimension n .

Proof.

We have the corresponding result for upper X Subscript n in Proposition 3.3.3. We can assume by induction that the result for the nested scheme holds for smaller values of n (for the base case note that upper X Subscript 0 comma 1 Baseline approximately-equals upper X 1 approximately-equals double-struck upper C squared right-parenthesis . Locally on a neighborhood of any point where upper P 1 comma period period period comma upper P Subscript n Baseline are not all equal, the result then follows from Lemma 3.5.5.

The locus where all the upper P Subscript i are equal is isomorphic to double-struck upper C Superscript 1 Baseline times upper Z where upper Z equals sigma Superscript negative 1 Baseline left-parenthesis ModifyingBelow 0 With bar right-parenthesis subset-of-or-equal-to upper H Subscript n minus 1 comma n is the zero fiber in the nested Hilbert scheme, the factor double-struck upper C Superscript 1 accounting for the choice of the common point upper P equals upper P 1 equals ellipsis equals upper P Subscript n on the x -axis upper V left-parenthesis y right-parenthesis subset-of-or-equal-to double-struck upper C squared . By a theorem of Cheah (Reference8, Theorem 3.3.3, part (5)) we have dimension upper Z equals n minus 1 .

3.6. Calculation of canonical line bundles

We will need to know the canonical sheaves omega on the smooth schemes upper H Subscript n and upper H Subscript n minus 1 comma n . To compute them we make use of the fact that invertible sheaves on a normal variety are isomorphic if they have isomorphic restrictions to an open set whose complement has codimension at least two.

Definition 3.6.1

Let z equals a x plus b y be a linear form in the variables x comma y . We denote by upper U Subscript z the open subset of upper H Subscript n consisting of ideals upper I for which z generates upper R slash upper I as an algebra over double-struck upper C . We also denote by upper U Subscript z the preimage of upper U Subscript z under the projection upper H Subscript n minus 1 comma n Baseline right-arrow upper H Subscript n