Lecture Notes in MathematicsEditors:A. Dold, HeidelbergF. Takens, GroningenB. Teissier, Paris

1721

SpringerBerlinHeidelbergNew YorkBarcelonaHong KongLondonMilanParisSingaporeTokyo

Anthony Iarrobino Vassil Kanev

Power Sums,Gorenstein Algebras,and Determinantal Loci

With an AppendixThe Gotzmann Theorems and the Hilbert Schemeby Anthony Iarrobino and Steven L. Kleiman

Springer

Authors

Anthony larrobinoMathematics DepartmentNortheastern UniversityBoston, MA 02115, USA

E-mail: iarrobin@neu.edu

VassiI KanevInstitute of MathematicsBulgarian Academy of Sciences1113 Sofia, Bulgaria

E-mail: kanev@math.bas.bg

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Die Deutsche Bibliothek - CIP-Einheitsaufnahme

Iarrobino, Anthony A.:Power sums, Gorenstein algebras, and determinantal loci / Anthonylarrobino ; Vassil Kanev. With an appendix The Gotzmann theoremsand the Hilbert scheme / by Anthony larrobino and Steven L.Kleiman. - Berlin; Heidelberg; New York; Barcelona; Hong Kong; London; Milan; Paris; Singapore; Tokyo: Springer, 1999

(Lecture notes in mathematics ; 1721)ISBN 3-540-66766-0

Mathematics Subject Classification (199 J):14M12, 14C05, 13C40, 14N99, 13HlO

ISSN 0075- 8434ISBN 3-540-66766-0 Springer-Verlag Berlin Heidelberg New York

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To our parents

Elizabeth and Anthony Iarrobino

Givka i Ivan Kanevi

-

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Preface

This book is devoted to a classical problem with a long history ­that of representing a homogeneous polynomial as a sum of powersof linear forms. This problem is closely related to another interestingtopic ­ the study of the loci which parametrize homogeneous poly­nomials with a given sequence of dimensions for the spaces spannedby their order­i higher partial derivatives. Here a convenient tool towork with are the catalecticant matrices associated to a homogeneouspolynomial, whose columns are the coefficients of its partial derivativesin appropriate monomial bases ­ the above dimensions are then theranks of the catalecticant matrices, and the above parametric varietiesare their determinantal loci.

In the Introduction we define all basic notions in an informal way,in the classical setting of characteristic zero. We hope this will fa­cilitate reading the book, where setting of arbitrary characteristic isadopted. Our experience has been that with a little more effort almostall results valid in characteristic zero can be extended to arbitrarycharacteristic, replacing the ring of polynomials by the ring of dividedpowers; the two rings are isomorphic when the characteristic is zero.

The first two chapters are mainly expository and are intended togive an account of what was already known about catalecticant ma­trices, especially those associated with the first partial derivatives orwith homogeneous polynomials in two variables. We aimed to makethis part of the book as self­contained as possible, and include fullproofs of some material scattered in the literature or contained insome hardly available old books. Chapters 3 ­ 8 as well as Section 2.2of Chapter 2 contain our new results on the subject. We also includedSections 4.4 and 6.4 in which some recent development due to vari­ous authors is surveyed ­ a development partially inspired by earlierpreliminary versions of this memoir circulated in 1995 ­ 1996 [IK].

The expert already familiar with the basic notions and notationmay wish to skip to the Brief Summary of Chapters at the end of theIntroduction, then skim the expository part of the first two chapters,

x

noting especially the Detailed Summary (Section 1.4), before lookingfor topics of particular interest.

ACKNOWLEDGMENT. We thank Mats Boij, Joel Briancon, YoungHyun Cho, Steve Kleiman, Philippe Maisonobe, Michel Merle,Juan Migliore, Yves Pitteloud, Richard Porter, Bernard Teissier,Junzo Watanabe, Jerzy Weyman, and Joachim Yameogo for theircomments. We particularly thank Anthony Geramita, Andy Kustin,Jan Kleppe, and Giuseppe Valla for detailed answers to our ques­tions, and Young Hyun Cho, Bae Eun Jung, Hal Schenck, and JunzoWatanabe for their careful reading of certain sections, and corrections.We are most appreciative to Steve Kleiman for joining with us inwriting Appendix C.

We are grateful to Richard Porter whose advice and LaTeX exper­tise guided us in improving the appearance of the book ­ and as wellfor access to an early version of his LaTeX guidebook [Por]; and wethank Tania Parhomenko for essential help in compiling the book inLaTex. The first author thanks the Laboratory of Mathematics at theUniversity of Nice and its members for their hospitality; and his spouseGail Charpentier for her support of the project. The authors were sup­ported in part by the Bulgarian foundation Scientific Research, andby the National Science Foundation under the US­Bulgarian projectAlgebra and Algebraic Geometry.

We thank Anthony and Elizabeth Iarrobino, parents of the firstauthor, for designing the frontispiece, based on a drawing by AnthonyIarrobino.

The authorsSeptember 1999

Contents

Introduction: Informal History and Brief Outline xiii0.1. Canonical forms, and catalecticant matrices of higher

partial derivatives of a form xiii0.2. Apolarity and Artinian Gorenstein algebras xviii0.3. Families of sets of points xxi0.4. Brief summary of chapters xxii

Part I. Cataleeticant Varieties 1

Chapter 1. Forms and Catalecticant Matrices 31.1. Apolarity and catalecticant varieties: the dimensions of

the vector spaces of higher partials 31.2. Determinantalloci of the first catalecticant, the Jacobian 161.3. Binary forms and Hankel matrices 221.4. Detailed summary and preparatory results 41

Chapter 2. Sums of Powers of Linear Forms, and GorensteinAlgebras 57

2.1. Waring's problem for general forms 572.2. Uniqueness of additive decompositions 622.3. The Gorenstein algebra of a homogeneous polynomial 67

Chapter 3. Tangent Spaces to Catalecticant Schemes 733.1. The tangent space to the determinantal scheme V s (u, v; r)

of the catalecticant matrix 733.2. The tangent space to the scheme Gor(T) parametrizing

forms with fixed dimensions of the partials 79

Chapter 4. The Locus PS(s, j; r) of Sums of Powers, andDeterminantal Loci of Catalecticant Matrices 91

4.1. The case r = 3 924.2. Sets of s points in JPT-l and Gorenstein ideals 102

XII

4.3. Gorenstein ideals whose lowest degree generators are acomplete intersection 108

4.4. The smoothness and dimension of the scheme Gor(T)when r = 3: a survey 116

Part II.Scheme

Catalecticant Varieties and the Punctual Hilbert129

Chapter 5. Forms and Zero-Dimensional Schemes I: Basic Results,and the Case r = 3 131

5.1. Annihilating scheme in IP'r-l of a form 1355.2. Flat families of zero-dimensional schemes and limit ideals 1425.3. Existence theorems for annihilating schemes when r = 3 1505.3.1. The generator and relation strata of the variety Gor(T)

parametrizing Gorenstein algebras 1515.3.2. The morphism from Gor(T): the case T (s, s, s) 1565.3.3. Morphism: the case T (s - a, s, s, S - a) 1675.3.4. Morphism: the case T (s - a, s, S - a) 1725.3.5. A dimension formula for the variety Gor(T) 1795.4. Power sum representations in three and more variables 1825.5. Betti strata of the punctual Hilbert scheme 1895.6. The length of a form, and the closure of the locus

PS(s, j; 3) of power sums 1975.7. Codimension three Gorenstein schemes in IP'n 201

Chapter 6. Forms and Zero-Dimensional Schemes, II: AnnihilatingSchemes and Reducible Gor(T) 207

6.1. Uniqueness of the annihilating scheme; closure ofPS(s, j; r) 208

6.2. Varieties Gor(T), T = T(j, r), with several components 2146.3. Other reducible varieties Gor(T) 2246.4. Locally Gorenstein annihilating schemes 226

Chapter 7. Connectedness and Components of the DeterminantalLocus IP'Vs(u , v ;r ) 237

7.1. Connectedness of IP'Vs(u , v; r) 2377.2. The irreducible components of Vs(u, v; r) 2417.3. Multisecant varieties of the Veronese variety 245

Chapter 8. Closures of the Variety Gor(T), and the ParameterSpace G(T) of Graded Algebras 249

Chapter 9. Questions and Problems 255

XIII

Appendix A. Divided Power Rings and Polynomial Rings 265

Appendix B. Height Three Gorenstein Ideals 271B.l. Pfaffian formulas 272B.2. Resolutions of height 3 Gorenstein ideals and their squares 276B.3. Resolutions of annihilating ideals of power sums 280BA. Maximum Betti numbers, given T 282

Appendix C. The Gotzmann Theorems and the Hilbert Scheme(Anthony Iarrobino and Steven L. Kleiman) 289

C.l. Order sequences and Macaulay's Theorem on Hilbertfunctions 290

C.2. Macaulay and Gotzmann polynomials 293C.3. Gotzmann's Persistence Theorem and m-Regularity 297C.4. The Hilbert scheme Hilb? (IP'r-l) 302C.5. Gorenstein sequences having a subsequence of maximal

growth, and Hilb? (IP'r-l) 307

Appendix D. Examples of "Macaulay" Scripts

Appendix E. Concordance with the 1996 Version

References

Index

Index of Names

Index of Notation

313

317

319

335

341

343

Introduction: Informal History and BriefOutline

0.1. Canonical forms, and catalecticant matrices of higherpartial derivatives of a form

A standard fact from linear algebra is that if f(x) = xAxt , X E Fis a quadratic form in r variables over a field k of characteristic not 2,then f can be represented as a sum of s squares of linear forms if andonly if the rank of A satisfies rk(A) ::; s. For homogeneous forms ofhigher degree one can ask a similar question.

PROBLEM 0.1. What are the conditions on a homogeneous poly­nomial (shortly a form) of degree j in r variables, so that it can berepresented as a sum

f = L{ + ... + (0.1.1)

where L, is a linear form and s is fixed? When is such a representationunique? If P S (s, j; r) (for power sum) denotes the set of such formsin the space R j of all degree­j forms, what are the generators of theideal of the affine variety P S (s, j; r)?

A particular case is Waring's problem for general forms

WARING'S PROBLEM. What is the minimum integer s such that ageneral form of degree j in r variables can be represented as sum ofpowers as in (0.1.1)?

Waring's problem was only recently solved by J. Alexander andA. Hirschowitz [AIH3]. Their result also yields (via Terracini's Lemma)the dimension of PS (s, j; r) (see Section 2.1 for details).

These problems attracted much attention among geometers andalgebraists who worked in the field of theory of invariants in the sec­ond half of the nineteenth and first decades of the twentieth century.It suffices to mention the names of A. Clebsch, J. Liiroth, T. Reye,J. J. Sylvester, G. Scorza, E. Lasker, H. W. Richmond, A. Dixon,

XVI Introduction,

A. Terracini, and J. Bronowski, who made important contributions tothe subject (see [Bra, Cle, Dix, Las, Lur, Rey, Ri, ScI, Sc2, Syl,Sy2, Sy3, Terl, Ter2]). When a representation of the type (0.1.1) isunique it is called a canonical form of f and a great deal of the abovecited research was devoted to finding canonical forms of homogeneouspolynomials. In fact even more general canonical forms close to sumsof powers were studied, as e.g. f = X 3+ y 3 + Z3 +mXYZ for cubicpolynomials in 3 variables (see R. Ehrenborg and G.-C. Rota's [EhR]for a modern account and amplification).

The case of two variables (binary forms) is not difficult and wasessentially solved by J. Sylvester [Syl, Sy2, Sy3] who proved thata general binary form of odd degree j = 2t - 1 has a canonical formf = + ... + He also introduced a catalecticant invariant inthe case of even degrees and proved that among the binary forms ofdegree j = 2t with vanishing catalecticant invariant every sufficientlygeneral one has a canonical form f = + ... + 1 An extensionof Sylvester's arguments is easy and the outcome is that the algebraicclosure PS(s, j; 2) is the rank :S s determinantallocus in the space of(s + 1) x (j - s+ 1) catalecticant matrices (to be introduced below); thecanonical forms are determined by the solutions of linear homogeneoussystems whose matrices of coefficients are catalecticant matrices 2.

If one wants to obtain canonical forms of arbitrary homogeneouspolynomials in 2 variables a new feature appears: one has to considerrepresentations more general than (0.1.1), namely

f = GILrdl+l + ... + (0.1.2)

where deg G, = d, - 1 and d, = s. We refer the reader toSection 1.3 for an exposition of this subject. A different solution tothe problem of representing a binary form as sum of powers of s linearforms was obtained by S. Gundelfinger [Gul, Gu2] (see also [GrY,

KuI, Ku2]), who expressed the condition f E PS(s, j; 2) in terms ofthe vanishing of certain covariants of f.

The problem of finding the generators of the ideal of PS (s, j; r)is more difficult. For instance in the case of quadrics (j = 2) this is

lSylvester introduced the name "catalecticant" from prosody, where a catalec-tic line of verse means one missing a foot, or beat: the catalectic binary forms havet summands, instead of the t+ 1 needed in general for binary forms of degree j = 2t(see B. Reznick's [Rezl, p. 49] for a historical remark).

2This should have been well-known to Sylvester and later invariant theorists,although maybe not written explicitly. At those times the aim was to express suchconditions in terms of invariants and covariants.

§ 0.1. CANONICAL FORMS AND CATALECTICANT MATRICES. .. XVII

the content of the Second Fundamental Theorem of invariant theoryfor the orthogonal group O(s) (see discussion in Section 1.2). In thebinary case (r = 2) the generators are the (s + 1) x (s + 1) minors ofcertain catalecticant matrices - the Hankel matrices. We refer to thepapers [GruP, Eil, Wa4] as well as to Section 1.3 about this result.

Much less is known if the number of variables r is greater thantwo. One of the primary goals of this book is to study the locusP S (s, j; r) (= P, for short) of homogeneous forms of degree j in rvariables which have a representation of the form (0.1.1). We call thisa length-s additive decomposition of f. We aim to generalize the abovementioned cases j = 2 or r = 2 and to relate the closure Ps to certaindeterminantal loci of catalecticant matrices, which we now introduceinformally. We refer the reader to Section 1.1 for precise definitions.

We will suppose that k is an algebraically closed field. Let R =

k[X1 , ... ,Xr ] denote the polynomial ring, and let R j be the space ofhomogeneous polynomials of degree j. Let us assume for the momentthat char(k) = O. If a form f E Rj has a length-s additive decompo-sition as in (0.1.1) (so f E PS(s,j;r)) then it is clear that for everyv, 1 :::; v :::; j - 1 the partial derivatives OV f /oXv with IVI = v spana subspace (L{-v, ... in R u, U = j - v of dimension no greaterthan s.

This can be equivalently stated in the following way. Consider thepolynomial ring R = k [Xl, ... ,xr ] and for fixed l, taking derivatives,consider the linear operator Cf (u, v) : R; -----t R u which transforms¢ E R; to ¢ 0 f = ¢( 01, ... ,Or)f. The above condition implies thatfor every v, 1:::; v:::; j -lone has rkCf(u,v) :::; s. In order to expressthis in matrix form take a basis VI + + vr = v for Rvand a basis (j-v)' x" .« (j-v)' X UI XUr U + + U = J' - v = U

U' Ull.·ur' 1 r , 1 r

for R u. Then an easy calculation shows that the matrix of Cf(u, v)for the polynomial

f =

.,J. XWI XWr

,

'

a W l , ... ,W r 1 ... rWI·" ·Wr ·WI+···+Wr=j

is equal to j!/(j - v)! times the catalecticant matrix

Catf(u, v; r) = (buy = au+v )lul=u,lVl=v' (0.1.3)

So, the above rank conditions can be equivalently stated by sayingthat the variety PS (s, j; r) is contained in the rank j, s determinantalloci of the catalecticant matrices (0.1.3) for each u, 1 :::; U :::; j - 1.Notice that tCatF(u,v;r) = CatF(v,u;r), so half of the values of usuffice, 1 < u :::; j /2.

XVIII Introduction, .

(0.1.4)

The catalecticant matrices in the binary case r = 2, which wediscussed above have the following form. If

f xt . Xj-IX (J) xirx: xi= ao 1 + Ja1 1 2 + ... + v av 1 2 + ... + aj 2'

one has with j = u + v

C ( 2) .a.::.1.) ,atf u,v; = ..

au au+l aj

which is also known as the u-th Hankel matrix of the binary form iThe Hankel matrices have been extensively studied because of appli-cations to control theory, to data-processing, or to discretization ofdifferential equations (see, e.g. [HeiR, 10, HMP, GoS]). Most ofthese studies have been in a non-homogeneous context; the homoge-neous context that we adopt may lead to more simply stated results.

In the general case we ask the following questions.

PROBLEM 0.2. What are the possible sequences of ranks of thecatalecticant matrices of a form f E PS (s1 j; r )? What kind of formsbelong to the boundary PS(s,j;r) - PS(s,j;r)?

In the binary case the second question is easy and requires consid-ering the decompositions (0.1.2) (this is attributed to J. H. Grace in[EhR]); the first question was solved by Macaulay [Mac1] (see alsoSection 1.3). We study this problem and in the case r = 3 we an-swer the first question, and we give partial results for the second inChapter 5 (see Section 5.6).

In general this problem as well as the question of the explicitforms of the homogeneous polynomials belonging to the boundaryPS( s, j; r ) - PS( s, j; r) (a generalization of the decomposition (0.1.2))are open problems. One can prove (see Lemma 1.17) that when f isgeneral- lies outside a certain proper closed subvariety of PS(s, j; r )- these ranks t, = rk Catf(i, j - i; r ) are equal to the following se-quence of positive integers T = (to, ... ,tj), ti = H( s, j, r )i, where

H(s,j, r)i = min(s, dirrn, tt; dim Rj-i), o:s; i < j. (0.1.5)

We conclude that PS(s,j; r) is contained in the algebraic closure of thequasiaffine set Gor(T) consisting of polynomials for which the ranksof the catalecticant matrices are equal to the sequence T = H(s,j,r)given by (0.1.5). Clearly Gor(T) is an open subset of the determinantal

§ 0.1. CANONICAL FORMS AND CATALECTICANT MATRICES .. XIX

locus given by the vanishing of all (s + 1) x (s + 1) minors of all thecatalecticant matrices Cat f (i, j - i; r) .

One of the main results in this book is that for r 3, j = 2t or2t+ 1, and s S dimk R t - 1 the affine variety PS(s,j; r) is an irreduciblecomponent of Gor(T) (Theorem 4.10A). Moreover, in case r = 3 onehas in fact equality (Theorems 4.1A and 4.5A). This result is a steptoward a solution of Problem 0.1 when r = 3. It can be reformulatedas follows. If s S Ct1) and a ternary form f of degree j = 2t or 2t + 1has the same sequence of ranks of its catalecticant matrices as that ofa general form of PS(s, j; 3), i.e. equal to (0.1.5), then f is a sum of spowers of linear forms (0.1.1) or is a degeneration of such a sum (i.e.belongs to PS(s,j;3)).

The sequence H(s,j,3) is only one of the possible sequences T =H f of ranks of catalecticant matrices for ternary forms f. In Chap­ter 5 we consider a more general and difficult problem ­ representing aternary form with arbitrary sequence T of ranks of catalecticant ma­trices as a sum of s powers of linear forms, or as a limit of such sums,where s = max{Td, the minimum possible number of summands.With the exception of some sequences occuring when the degree j 4is even, we find criteria for such a representation to exist, in terms ofthe catalecticant matrices. While the proofs are complicated and usethe Buchsbaum­Eisenbud structure theorem for height three Goren­stein ideals, from commutative algebra, the criteria are simple and areanalogous to the rank conditions for Hankel matrices in the binarycase. In particular, if the form f is such that the sequence T of ranksof catalecticant matrices contains a constant subsequence (s, s, s), thenf is in the closure PS(s,j, 3). The reader may find a further discussionand a precise statement of our results in this direction in the summa­rizing Section 5.4, which is written in the language of matrices andpolynomials, so as to be accessible to a general reader.

The geometrically minded reader might look at this subject fromanother point of view. The projectivization lP'PS( s, j; r) is the s­secantvariety of the Veronese variety Vj(lP'T-l). This is the closure in lP'T­lof the variety traced out by the projective subspaces spanned by thes­tuples of points of Vj(lP'T-l). So the above results can be restatedin the following way. Provided s S dirrn, Rt - 1 as above, the s­secantvariety to the Veronese variety Vj (lP'T­l) is an irreducible component ofthe determinantal locus given by the vanishing of the (s + 1) x (s + 1)minors of all catalecticant matrices associated to forms of degree j.A warning: lP'Gor(T) need not be equal to the latter determinantalvariety (see Example 7.11). In some cases (j = 2, or r = 2, or s S 2) it

xx Introduction, ...

is known that the (s+ 1) x (s+ 1) minors of the catalecticant matricesgenerate the homogeneous ideal of the s-secant variety to the Veronesevariety (equal to the ideal of PS(s,j;r), d. Problem 0.1). We refer toSections 1.2 and 1.3 for a discussion.

0.2. Apolarity and Artinian Gorenstein algebras

The notation Gor(T) comes from another important connection,the study of graded Artinian Gorenstein algebras. Setting Xi = Oi wemay think of R as the ring of linear differential operators with constantcoefficients acting on the ring R of polynomials. Let f E R j , f i- 0,and let I c R be the ideal of all differential operators which annihilatef; these are also called polynomials apolar to f if considering R =

k[Xl,' .. ,xr ] as before. We use the notation I = Ann(f). Theseideals were called principal systems by F. H. S. Macaulay and thequotients A f = R/I were studied in [Mac2]. In fact Macaulay workedover arbitrary characteristic of the base field, which requires a slightlydifferent setting, which is equivalent (if char(k) = °or char(k) > j) tothe one we have so far considered (see page 266).

It is well known that the quotients Af are exactly the graded Artinalgebras A whose socle,

Soc(A) = °:m = {h E A 1m· h = 0, where m = R?l = R 1 + ... }satisfies dims Soc(A) = 1 (see e.g. [Ei2, p.526], or also Lemma 2.14 be-low). These algebras Af = R/ Ann(f), known classically and studiedextensively by Macaulay and others, became known as graded ArtinianGorenstein algebras, after the influential article of H. Bass [Bas]3:D. Gorenstein had studied a self-duality property of semigroup ringsfor certain singularities of curves [Gor]. We will follow usual practicein calling 1= Ann(f) a Gorenstein ideal- although it is the quotientalgebra A f that is Gorenstein and that in the Artinian case has finitedimension as a k-vector space.

Recall that the Hilbert function of a graded algebra A = EBA is thesequence H = (ho, . . . ,hi," .), where hi = dirnj, Ai. The dimensionsof the spaces of partial derivatives of f (see (0.1.3)) form a sequenceH f equal to the Hilbert function H(A f ). From this point of viewit is natural to consider arbitrary sequences of positive integers T =(to, ... ,tj) with to = 1, tl ::; r, ti = tj-i and let Gor(T) denote thesubset of the affine space A j = A(Rj) consisting of the polynomials fwith T = H(A f ) . Then if Gor(T) is nonempty, lP'Gor(T) parametrizesall graded Artinian Gorenstein algebras A f with Hilbert function H f =

3R. Bass reports that the name was originally given by A. Grothendieck

§ 0.2. ApOLARITY AND ARTINIAN GORENSTEIN ALGEBRAS XXI

H(A j ) = T. We call such a sequence T a Gorenstein sequence, andwe let H(j, r) be the set of all such Gorenstein sequences.

When r = 3 a powerful tool that we and other authors have usedfor studying the varieties of sums of powers PS( s, j; 3) and the varietiesGor(T) is the Buchsbaum-Eisenbud structure theorem of height threeGorenstein ideals [BE2]. It gives in particular the minimal resolutionof the ideal Ann(J), for f E k[X, Y, Z] in terms of a skew-symmetricmatrix and its Pfaffians. From this it is straightforward to determinethe set of height three Gorenstein sequences H(j,3) (see [StL], andSection 4.4 below.)

We consider several related problems concerning the decompositionof Aj = A(Rj) as a disjoint union

A j = U Gor(T).TEH(j,r)

The questions we study or survey are

PROBLEM (A). Determine the minimum length s of an additive

decomposition f = L{ + ... + of a general degree-j homogeneouspolynomial f in r variables as sums of s powers of linear forms (War-ing's problem). When is there a unique additive decomposition ofminimum length? How is the Zariski closure PS(s,j,r) in A(Rj ) ofthe variety PS(s, j; r) parametrizing forms having an additive decom-position of length s, related to the determinantal loci of catalecticantmatrices? What is the dimension of P S (s, j; r)?

PROBLEM (B). Describe the algebraic set Gor(T) C A(Rj ) . IsGor(T) irreducible? What is its dimension? What is its Zariski clo-sure?

PROBLEM (C). If T is a given sequence, and 1 = Ann(J) is aGorenstein ideal in R such that A f = R/1 has Hilbert function H (A j) =

T, determine the possible Hilbert functions and minimal resolutionsfor the ideal 12 .

PROBLEM (D). Given a set Z of s general points of pr-l, consider

the square and the symbolic square of the graded ideal Iz inR = k[Xl' ... ,xr ] of polynomials vanishing at Z. What are the Hilbert

functions of and R/I12 )?

PROBLEM (E). Generalizing the binary forms case r = 2, under-stand the relation between the Gorenstein ideal 1 = Ann(J)' f EGor(T), and the ideal J = generated by the elements of 1 havingdegree no greater than c, especially when c :::::! j /2 and J is the defining

XXII Introduction, .

ideal of a zero-dimensional (or punctual) scheme in IF'r-I. What is theconnection between Gor(T) and the punctual Hilbert scheme?

Problems (A) and (B) we discussed above. As we show in Sec-tions 3.2 and 4.1 Problem (C) is related to the study of the tangentspaces to the determinantal scheme Gor(T) whose associated reducedscheme is the locus Gor(T) considered above. As for Problem (D), whydo ideals Iz of finite sets of points appear in the problem of represent-ing a homogeneous form f as sums of powers of linear forms? This isthe core of the classical apolarity method, which we use throughout inthe memoir, and connecting 12 with and eventually with per-mits us to obtain some results on Problems (A) and (B). We introducethe reader to it by sketching the beautiful solution of Clebsch [Cle] tothe problem of representing a quartic form in 3 variables as a sum of4th powers of 5 linear forms. If one tries to represent such a form assum of powers

f = Li+ ... + Lg (0.2.1)

the first thought is that this should always be possible for a genericf E R 4 since on both sides we have the same number of parameters,

3 -215. Now, let L: = Lj=l PijXj, let Pi = (Pi1 : Pi2 : Pi3) ElF'. LetR = k[X1' X2, X3]. We may think of R as the homogeneous coordinatering of the dual projective space p2, where 1F'2 has homogeneous coor-dinate ring R = k[X1, X 2, X 3]. This is natural by the differentiationpairing. Now, consider the five points PI, ... ,P5 E p2 which corre-spond to the five lines L, = 0 in (0.2.1). There is a quadratic form1; E R2 which vanishes simultaneously on all Pi· Differentiate f by1;(01,02,03), Then an easy calculation (see e.g. Lemma 1.15) showsthat

Equivalently, the linear map Cf(2, 2) : R2 ----7 R2 whose matrix is thecatalecticant matrix Catf(2, 2; 3) has a nonzero element 1; in the ker-nel, hence the catalecticant determinant det(Catf(2, 2; 3)) = O. This isa polynomial relation of degree 6 on the coefficients of f in order thatf has a representation of the form (0.2.1). Why is this a nontrivialrelation? Take 9= Li+ ... +Lt with 6 general forms. Then the sameargument as above coupled with the fact that no conic in p2 can passthrough 6 general points, shows that Catg(2, 2; 3) is nondegenerate.Thus det(Catg(2, 2; 3)) i- O. In fact, one can show that the hyper-surface of degree 6 in A15 given by the vanishing of the catalecticant

§ 0.3. FAMILIES OF SETS OF POINTS XXIII

determinant is equal to the variety PS(5, 4; 3) (see e.g. [DK, §6] orCorollary 2.3).

Remark. All problems and discussion so far were stated for sums ofpowers. In fact we work mainly with another type of additive decom­position 1 = Lfl + ... + LV] which permits us to obtain new results,and extend many old ones to arbitrary characteristic of the base fieldk. Namely, 1 belongs to the ring of divided powers V and LIJ] is thedivided power of a linear form L. We refer to Appendix A for defi­nitions and details. The reader who is interested in the char(k) = 0case or characteristic sufficiently large (char(k) > i. when workingwith polynomials of degree j, j) may replace the ring of divided pow­ers V by the ring of polynomials R: and the divided powers L[jl byordinary powers Lj in all statements of the book ­ since under theseassumptions LIJ] = (see Proposition A.12).

J.

0.3. Families of sets of points

A representation of 1 as a sum of powers of linear forms as in (0.1.1)determines s distinct points of lP'n, or a point of the s-th symmetricproduct SymS(lP'n), ti = r - 1. Likewise, a form 1 in the closure ofPS(s,j;r), may determine a degree­s zero­dimensional subscheme Zjof lP'n, that is smoothable ­ lies in a fiat family of such schemes, thegeneral member of which consists of s distinct points, so is smooth.

A degree-s, zero­dimensional subscheme Z of lP'n consists of s points,counting multiplicities, with the further structure of an Artin ringconcentrated at each distinct point ­ these are sometimes called thickpoints. Such a scheme Z determines a point of the punctual Hilbertscheme Hilbs(lP'n) , introduced by Grothendieck, and studied furtherby the Nice school of J. Briancon, M. Granger, J. Yameogo, as wellas J. Fogarty, S. Kleiman, and others. Recently the punctual Hilbertscheme, particularly for surfaces, has seen striking applications to sym­metric functions and physics (see, for example [Hai, Nak l , Nak2]).When ti ::; 2, Hilb" (lP'n) is smooth. This fact has been recently used byG. Ellingsrud and A. Stromme to determine the degree of the secantvariety to the j-th Veronese embedding of lP'2 in certain cases [EIS].

How does a zero­dimensional scheme Z C lP'n connect to a form17 Given Z, we can consider the set of forms 1 of degree-j, such thatIz C 1= Ann(j). If we are very fortunate, this process determines avector bundle of forms 1 over Hilb{P"}. However, this is not alwaysthe case, and the study of when there might be a fibration Gor(T)to H'ilb" (lP'n) is one of the more technically difficult problems that weexplore in Part II.

XXIV Introduction,

0.4. Brief summary of chapters

We now briefly describe the contents of the book, and postponea more detailed summary of the main results until Section 1.4. Forsimplicity we state the results in the framework of usual power sums,which are valid for char(k) = 0 or char k > j (sufficiently large),although the theorems in the paper are stated and proved for dividedpowers and arbitrary characteristic (see the remark above).

Chapter 1 contains preliminary material, examples and definitions,a discussion of determinantal varieties of catalecticant matrices in thesimplest cases Cat(l,j - l;r) and Cat(i,j - i;2) (Sections 1.2, 1.3),and the detailed summary (Section 1.4). In Section 1.1 we define thedeterminantal scheme V s (u, v; r) of catalecticant matrices, whose idealis generated by the (s+ 1) x (s+ 1) minors of the generic catalecticantmatrix CatF(u, v; r). We also define the scheme Gor(T) whose asso-ciated reduced subscheme Gor(T) we discussed above. We introducethe basic definitions of apolarity, prove the Apolarity Lemma 1.15 andgive some corollaries of it (cf. [DK, §2,4] for this classical material).

Section 1.2 is a survey of the known results about the determinantallocus Vs (1, j - 1; r), where s < r. Besides the classical case of quadrics(j = 2) we describe a recent work of O. Porras [Po] about this varietyand report on a recent result of the second author [Ka] describingthe structure of PS(2, j; r), as well as that of Gor(T) when tl ::; 2.Section 1.3 is a self-contained exposition with full proofs of the theoryin the binary forms case (r = 2). It may serve as an introductionto the subject, and as a model for what we would like to achieve forr 3. The detailed summary, Section 1.4, also states results from theliterature that we use in the sequel.

Chapter 2 contains a discussion of the Waring; problem (Section 2.1),a new result concerning the uniqueness of the representation of a formf as sum of powers (Section 2.2), and an introduction to ArtinianGorenstein algebras (Section 2.3). Section 2.1 contains a translationof the J. Alexander-A.Hirschowitz vanishing theorem to a solutionof the Waring problem, in all cases except when char klj; and Sec-tion 2.2 contains Theorem 2.6, which proves that a representationf = L{ + ... + is unique provided f E PS (s, j; r) is general enoughand s ::; dimj, Rt - r for j = 2t, or s ::; dimj, Rt for j = 2t + 1.

Section 2.3 is an introduction to Artinian Gorenstein algebras: itincludes all the basic results needed for the sequel. It also includes anapplication of the Minimal Resolution Theorem for ideals of s generalenough points of p2, to determine the minimal resolution of Ann(f),

§ 0.4. BRIEF SUMMARY OF CHAPTERS x:xv

when r = 3 (Proposition 2.19): this is an easy example of the relationbetween the study of points in jpr-l and the study of P S (s, j; r), thatis deepened in the later chapters of the book.

Chapter 3 contains two simple but important results, Theorems 3.2and 3.9 which reduce the calculation of the tangent spaces to theschemes Vs(u, v; r) and Gor(T) at a point i, to determining thedegree-j graded piece of the square 12 of the annihilating ideal 1 =Ann(J). We give also several examples and corollaries, including aproof of the irreducibility of the catalecticant determinantdet(CatF(t, t; r)) in Proposition 3.13.

More generally, in Theorem 3.14 and Remark 3.15 we give a re-sult which describes the structure of the corank-one determinantalloci Vs(u,v;r) with «< v, and s = dimkRu -1. Lemma 3.17 pro-

vides the connection between 12 and the symbolic square I12) , where

f = L{ + ... + and Pi E pr-l are the points in the dual space whichcorrespond to L i , and Z = {Pl,' .. Ps}. This lemma is the basic toolfor the calculation in Chapter 4 of the tangent space to Gor(T) forT = H(s, j; r) and s in the range s ::; dirrn, Rt - 1 , j = 2t or 2t + 1.Chapter 3 concludes with three conjectures. Conjecture 3.25 concernsthe Hilbert function of It is related to Conjectures 3.20 and 3.23about Gor(T), T = H(s, j; r), when s is in the range dirrn, Rt - 1 < s <dirrn, Rt , j = 2t or j = 2t + 1.

Chapter 4 contains some of the main results of the book. In The-orems 4.lOA and 4.10B we prove that, provided s ::; dirrn, Rt- 1 withj = 2t or 2t + 1, then the closure PS (s, j; r) is an irreducible compo-nent of both Gor(T), T = H(s,j, r) and of the determinantal locusVs (t, j - t; r). When r = 3, by S. J. Diesel's result showing the irre-

ducibility of Gor(T) for any Gorenstein sequence [Di], it follows thatin fact PS(s, j; r) = Gor(T).

Examples in Chapter 7 show that Vs(t,j - t;r) may be reduciblewhen r = 3 and Gor(T) can be reducible for certain T when r 2: 4, soan equality PS(s,j;r) = Gor(T), with T = H(s,j,r) seems to holdonly for r ::; 3. For s in the range dimj, Rt - 1 < s < dimj, R; thedimension of Gor(T) is larger than the dimension of PS(s, j; r). Insome cases we are able to prove that the scheme Gor(T) is genericallysmooth along PS(s, j; r), and that the unique irreducible component ofGor(T) which contains PS(s, j; r) is of expected dimension (Theorem4.1B for r = 3, j = 2t; Theorem 4.13 for r 2: 3, s > dimk(Rt) - r).

XXVI Introduction, ...

In Section 4.3, we study families of Gorenstein algebras R/Ann(f)such that the initial generators of Ann(f) form a complete intersec­tion: we show that they form an irreducible component of Gor(T)(Theorem 4.19). This work is related to results of J. Watanabe in[Wa2].

Our results in this chapter were obtained in 1994­1995. Afterdistributing the preliminary draft of the early version of this memoir[IK], our Theorems 4.1A, 4.1B, 4.5A, 4.5B for r = 3 were reprovedand substantially generalized by various authors. They use our Theo­rem 3.9 and some techniques specific to the r = 3 case, in particularthe Buchsbaum­Eisenbud structure theorem for height 3 Gorensteinideals. At present, it is proved that for r = 3, the schemes Gor(T) aresmooth for every Gorenstein sequence T (J. Kleppe, [KI2] ) and thereare several general formulas for dim Gor(T) in terms of T. We surveythese developments in Section 4.4 (see also [19] for a broader surveythrough 1997). We think it is worthwhile to include our original argu­ments since they focus more on apolarity and the ideals associated tofinite sets of points, so they seem more promising for generalization toarbitrary r.

In Chapter 5 we consider mainly forms in 3 variables. We focus onthe problem of finding criteria for representing a ternary form belong­ing to Gor(T) as a sum of s powers of linear forms, where s = max{Tdis the minimum possible number of summands. We also study theproblem of explicitly describing the boundary PS(s, j; r) ­ PS(s, j; r),obtaining some partial results when r = 3. The case of binary formssuggests that in order to obtain satisfactory results in these and otherrelated problems it is desirable to extend the classical apolarity rela­tion between finite sets of points Z in ]ji>r­l and forms, to one betweenzero­dimensional subschemes of ]ji>r­l and forms. This is done in Sec­tion 5.1 where we introduce and give some basic properties of annihi­lating schemes of a form. These are zero­dimensional schemes, whosegraded ideal consists of polynomials ¢ which annihilate f: ¢o f = O.For a f E Gor(T) such a scheme is called tight if deg Z = s = max{Td.Section 5.2 contains some material about limits of punctual schemesand limits of ideals which is used later in connection with studyingthe closure PS( s, j; r). We also give the definition and some properiesof the "postulation" Hilbert scheme HilbHpr­l whose closed pointscorrespond to zero­dimensional subschemes of pr­l with fixed Hilbertfunction.

Section 5.3 contains our results on power sum representations whenr = 3. In particular we show that if a ternary form f of degree j 2: 4

§ 0.4. BRIEF SUMMARY OF CHAPTERS XXVII

is such that the sequence T = H f of ranks of catalecticant matrices(recall that (H f) i is the dimension of the vector space of order j - ihigher partial derivatives of f) contains a subsequence (s, s, s), thenf has an annihilating scheme Zf of degree s. Moreover this schemeis unique, and its graded ideal is generated by the forms ¢ of degree::; T + 1 apolar to t, ¢ 0 f = 0, where T = min{i IT; = s}; so Zf canbe recovered explicitly from f. When f E Gor(T) is general enoughthe annihilating scheme is smooth - a set of distinct points in JP>2- and one obtains by apolarity a length-s power sum decompositionf = Notice the complete analogy of this result with thebinary case, where a similar statement holds when T (s, s). If Tdoes not contain (s, s, s) it is no longer true that a general elementf E Gor(T) has a length-s power sum representation.

When j = deg(f) is odd (so T (s, s)), we give necessary andsufficient conditions for the existence of annihilating schemes of degrees, and again they are unique and can be explicitly recovered from f, asabove. Our results in the even case T c (s - a, s, s - a), a> 0 are lesssatisfactory, and our methods do not cover some classes of sequencesT: however, these are exactly the cases when even if an annihilatingscheme of degree s existed, its ideal could not be generated by apolarforms of degree j, j /2, so the above procedure to recover Zf explicitlyfrom f could not be applied.

The canonical wayan annihilating scheme is associated with aform permits us to construct, when T (s, s, s) a morphism fromGor(T) to the postulation stratum HilbH lF'2 = (HilbH

lF'2) r ed (Theo-rem 5.31). The morphism is dominant and fibered by open subsetsin AS. As an application one obtains a dimension formula for Gor(T)when T (s, s, s), derived from Gotzmann's formula [Got3, Got5]for dimHilbH lF'2 (Corollary 5.50). Conversely, a simple formula fordim HilbH lF'2 , can be derived from the recent Conca-Valla formula[CoVI] for dim Gor(T) (Corollary 5.34). In Corollary 5.49 it is provedthat the Gorenstein ideals Ann(f) satisfy a certain weak Lefshetz prop-erty, when T (s, s, s).

Section 5.4 summarizes some of the results from the previous sec-tions. Here we avoid the language of commutative algebra used inSection 5.3, and formulate our results in terms of matrices and poly-nomials. We focus on how to explicitly calculate a power sum rep-resentation of a given form. An algorithm and some examples aregiven. In Section 5.5 some further applications of the morphism p :Gor(T) ---+ HilbH p 2 are given. Using it we derive some results aboutthe Betti strata of H ilbH lF'2 - the locally closed subsets of H ilbH (IF'2)

XXVIII Introduction, .

parametrizing schemes Z whose ideal sheaves have fixed generator de­grees.

In Section 5.6 we apply our work towards answering Problem 0.2above, for r = 3: we determine the set of Hilbert functions that occurfor forms f in the closure PS( s, j; 3) (Proposition 5.70). Also, it followsfrom the results of Section 5.3, that the sets Gor(T), T =::l (s, s, s)are completely inside the closure of PS(s,.i; 3) (Theorem 5.71). Wegive other criteria, that test either f E PS(s,.i;3) or the contrary.Nevertheless when r 2:: 3 the problems of describing explicitly theforms in the boundary PS (s,i, r) ­ PS (s, j; r), and of finding effective,simple criteria in terms of the form f itself, to decide whether each fis in PS(s,j;r), remain open, even when r = 3.

In Section 5.7 we discuss a generalization to higher dimensions ofthe apolarity relation between punctual schemes and graded ArtinianGorenstein algebras.

In Chapter 6 we study the component structure of Gor(T), whenr 2:: 4, extending our work on the annihilating scheme. We workout the example where the annihilating scheme consists of one point,and its defining ideal in the local ring of the point is equal to theannihilating ideal of some homogeneous polynomial in r ­ 1 variables.We call this a "conic Gorenstein scheme" concentrated at a singlepoint. In Lemma 6.1 we prove several statements that give a parallelbetween the case Z is smooth (this is the usual apolarity), and theconic Gorenstein case.

A main application of our tight annihilating schemes ­ thosewhose degree equals s = max{Td ­ is the construction of familiesGor(T) having two irreducible components for r 2:: 5, r i= 8 variables.This contrasts with the irreducibility of Gor(T) for r = 3, proven byS. J. Diesel [Di]. The cases r = 5,6 are treated in Corollaries 6.28 and6.29 and require j relatively high. For r 2:: 7, r i= 8 we consider the se­quence T = T(), r) where T(j, r) = (1, r, 2r ­1, 2r, . . . ,2r, 2r ­1, r, 1).In Theorem 6.26 we prove under the assumption j 2:: 8, r i= 8 thatGor(T), T = T(j, r) has at least two irreducible components. One ofthem, 0 1(T) has a Zariski open subset which consists of polynomi­

als f = L{ + ... + where the points PI, ... ,P2r E lPr - 1 whichcorrespond to the linear forms Li, are a self­associated set Z (see[Co2, Col, DO]). The other component 02(T) consists of poly­nomials whose tight annihilating scheme is a conic Gorenstein schemeconcentrated in one point, and whose ideal at the point is the annihi­lating ideal of a general cubic form g in r ­ 1 variables.

§ 0.4. BRIEF SUMMARY OF CHAPTERS XXIX

For any r , we show that the forms having a smoothable tight an­nihilating scheme ­ of degree s = maxi {(H f)i} - are in the closureof PS(s,j; r ); but we prove a weaker statement than the converse(Proposition 6.7). Nevertheless, this approach suffices to give exam­ples of reducible Gor(T), for r 2': 5, r i- 8.

We should notice that the cases r = 4, r = 8 were settled byM. Boij [Bo2], who constructed two irreducible components of certainGor(T); his construction works for arbitrary r 2': 4 and sufficientlylarge degrees of the forms. In Section 6.4 we discuss this and otherrecent developments, in the construction of components of Gor(T).

In Chapter 7 we first show that the projectivization IP'Vs (U, v; r )is connected: in fact, each component contains lP'PS (1, j; r ) (Theo­rem 7.6). We then in Section 7.2 give several criteria for deciding iff E Gor(T); and we use the criteria to show that even when r = 3,Vs(t, t; 3) may have a large number ltj4J irreducible components.

In Section 7.3 we restate some of the results obtained for P S (s, j; r)in terms of the s­secant variety to the Veronese variety S ec; (Vj (lP'r­l ));one has lP'(PS(s,j;r)) = Secs(Vj(lP'r-l)). We also report a result byEllingsrud­Strernme which gives the degree of Secs (lP'2) when s ::::; 8.

Chapter 8 compares the structure and closure of two natural em­beddings of lP'Gor(T): the first, into lP'(Dj) , is the one that we usethroughout; the second one sends an ideal into the ordered set of itsgraded components, in a product of Grassmanians.

In Chapter 9 we discuss open problems.Appendix A contains the definition of the ring of divided powers

and gives an exposition of the few properties of these rings we use inthe book. In particular Proposition A.12 shows that when char(k) = 0or char(k) is sufficiently large one may interchange the ring of dividedpowers D and the polynomial ring R.

Appendix B contains several results, mostly concerning heightthree Gorenstein algebras, that we use. Section B.1 contains basicdefinitions concerning Pfaffians, and as well a result con cering thePfaffian of a matrix with a large block of zeroes; and a decompositionformula of H. Srinivasan for Pfaffians. Section B.2 states for referencethe Buchsbaum­Eisenbud structure theorem for height three Goren­stein ideals, in the graded case we use; and a result of Kustin­Ulrichconcerning the minimal resolution of the square 12 when 1 is a heightthree Gorenstein ideal. Section B.3 states several useful results ofM. Boij connecting the minimal resolutions of the defining ideal forpunctual schemes of lP'n, and the minimal resolutions for related Ar­tinian Gorenstein algebras. In Section B.4 we state and prove a nice

xxx Introduction, ...

formula of A. Conca and G. Valla determining the maximal set of gen­erator degrees Dmax(T) possible for a height three Gorenstein sequenceT, directly from the second difference of T. We also show a result ofS. J. Diesel enumerating height three Gorenstein sequences T of givenorder and socle degree, by showing a one­to­one correspondence be­tween the maximal sets of generator degrees, and certain partitionsP(T).

Appendix C was written by the first author and Steven L. Kleiman,and it holds a particular place in the book. Section C.1 contains whatis used in the body: Macaulay's characterization of Hilbert functionsof graded quotients of the polynomial ring, using O­sequences, andthe particular case of graded quotients of k[Xl' X2] (Corollary C.6).The rest of the appendix treats various aspects of the theory of theHilbert scheme of HilbP (lP'n), which parametrizes subschemes of IP'nwith Hilbert polynomial P. The last Section C.5 proves some newresults about annihilating schemes of homogeneous polynomials andthe varieties Gor(T).

In Section C.2 we define Macaulay and Gotzmann polynomials,and give some of their properties; in Section C.3 we state the Per­sistence and Regularity Theorems of G. Gotzmann [Gotl]. Next, inSection C.4 we report on the relation of these results to G. Gotzmann'sidentification of the Hilbert scheme HilbP (lP'n) as a simply describedsubscheme in a product of two Grassmanians. Gotzmann's descriptionis based on Grothendieck construction and on his own improvementof Mumford's effective bound for the Castelnuovo­Mumford regularitydegree. Here we also establish D. Bayer's related conjecture that theHilbert scheme is a determinantal subscheme of a Grassmannian.

The last Section C.5 of the Appendix applies the Gotzmann HilbertScheme Theorem and a related result [Got4] to certain schemes Gor(T),in a manner similar to some applications in an article of A. Bigatti,A. Geramita, and J. Migliore [BGM]. Thus we obtain another ex­ample, in addition to those of M. Boij [Bo2], of a scheme Gor(T) forr = 4 having several irreducible components.

In Appendix D we give some examples of the Macaulay computeralgebra scripts used [BaSl].

In Appendix E we give a comparison with the earlier manuscriptof this book of May, 1996 [IK]. We do this in particular because somepublished articles refer to this earlier version, which was circulated.

Throughout the book we assume, unless otherwise specified, thatthe base field k is algebraically closed of arbitrary characteristic. Wemean by variety a separated, reduced scheme of finite type over k (i.e.

§ 0.4. BRIEF SUMMARY OF CHAPTERS XXXI

not necessarily irreducible). Unless otherwise stated, by a point of ascheme we mean a closed point of a scheme. We will use as a rule mathbold font to denote schemes and math slanted font to denote the corre­sponding varieties ­ the associated reduced subschemes. For instancewe frequently work with the schemes Gor(T), Vs(u, Vi r) etc. and thecorresponding reduced subschemes (varieties) Gor(T), Vs (u, Vi r) etc.Expressions like "Property 'P' holds for sufficiently general (or generalenough, or general) element 1" mean that from the context it is clearthat f belongs to an irreducible variety and Property 'P' holds forevery f in a certain Zariski open dense subset of this irreducible vari­ety. The expression ' ...for generic F ...' means F is the scheme­genericpoint of the irreducible scheme considered.

Dependence of the chapters/sections. A reader new to thismaterial or to our viewpoint on it might wish to start with this In­troduction and Sections 1.1­1.3, to get a flavor of the book, then lookthrough 1.4 and Chapter 2, before perusing later sections of interest.

Prerequisites:Each Chapter may be read independently, after reading Section 1.1.

Although Chapter 4 uses Chapter 3, it is not necessary to read Chap­ter 3 first, just look at what one uses. We do feel that Chapters 1and 2 give a desirable foundation for further reading. Here are somefurther suggestions.

Chapter 3: 1.1, 2.3; Section 3.2: 3.1.Section 4.2: 4.1Chapter 5: 1.1, 2.3; Section 5.3: 5.1, 1.3; Section 5.4: 1.1 only;

Section 5.5: 5.2; Section 5.6: 5.1,5.2; Section 5.7: 5.3Chapter 6: 5.1,5.2; Section 6.3: 6.2Chapter 7: 5.1; Section 7.2: 7.1Chapter 8: 2.3.