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This is page v Printer: Opaque this Contents Preface: Algebra and Geometry What Are Syzygies? The Geometric Content of Syzygies What Does Solving Linear Equations Mean? Experiment and Computation What’s In This Book? Prerequisites How Did This Book Come About? Other Books Thanks Notation ix ix xi xi xii xiii xv xv xv xv xvi Free Resolutions and Hilbert Functions The Generation of Invariants Enter Hilbert 1A The Study of Syzygies The Hilbert Function Becomes Polynomial 1B Minimal Free Resolutions Describing Resolutions: Betti Diagrams Properties of the Graded Betti Numbers The Information in the Hilbert Function 1C Exercises 1 2 9 First Examples of Free Resolutions 2A Monomial Ideals and Simplicial Complexes Simplicial Complexes Labeling by Monomials Syzygies of Monomial Ideals 13 13 13 14 16 vi Contents 2B 2C 2D Bounds on Betti Numbers and Proof of Hilbert’s Geometry from Syzygies: Seven Points in P The Hilbert Polynomial and Function and Other Information in the Resolution Exercises Syzygy Theorem 18 20 20 22 24 29 30 37 40 44 51 51 54 60 62 63 Regularity of Projective Curves A General Regularity Conjecture Proof of the Gruson–Lazarsfeld–Peskine Theorem Exercises 67 67 69 79 Points in P 3A The Ideal of a Finite Set of Points 3B Examples 3C Existence of Sets of Points with Given 3D Exercises Invariants Castelnuovo–Mumford Regularity 4A Definition and First Applications 4B Characterizations of Regularity: Cohomology 4C The Regularity of a Cohen–Macaulay Module 4D The Regularity of a Coherent Sheaf 4E Exercises The 5A 5B 5C Linear Series and 1-Generic Matrices 6A Rational Normal Curves 6A.1 Where’d That Matrix Come From? 6B 1-Generic Matrices 6C Linear Series 6D Elliptic Normal Curves 6E Exercises 81 82 82 84 86 94 103 Linear Complexes and the Linear Syzygy Theorem 7A Linear Syzygies 7B The Bernstein–Gelfand–Gelfand Correspondence 7C Exterior Minors and Annihilators 7D Proof of the Linear Syzygy Theorem 7E More about the Exterior Algebra and BGG 7F Exercises 109 110 114 119 124 125 131 Curves of High Degree 8A The Cohen–Macaulay Property 8A.1 The Restricted Tautological 8B Strands of the Resolution 8B.1 The Cubic Strand 8B.2 The Quadratic Strand 8C Conjectures and Problems 8D Exercises 135 136 138 142 144 148 157 159 Bundle Contents vii Clifford Index and Canonical Embedding 9A The Cohen–Macaulay Property and the Clifford Index 9B Green’s Conjecture 9C Exercises 165 165 168 172 Appendix Introduction to Local Cohomology A1A Definitions and Tools A1B Local Cohomology and Sheaf Cohomology A1C Vanishing and Nonvanishing Theorems A1D Exercises 175 175 182 185 186 Appendix A Jog Through Commutative Algebra A2A Associated Primes and Primary Decomposition A2B Dimension and Depth A2C Projective Dimension and Regular Local Rings A2D Normalization: Resolution of Singularities for Curves A2E The Cohen–Macaulay Property A2F The Koszul Complex A2G Fitting Ideals and Other Determinantal Ideals A2H The Eagon–Northcott Complex and Scrolls 189 190 193 195 197 200 204 207 209 References 215 Index 224 This is page viii Printer: Opaque this This is page ix Printer: Opaque this Preface: Algebra and Geometry ✂✁☎✄☎✆✞✝ ✟ Syzygy [from] Gr yoke, pair, copulation, conjunction — Oxford English Dictionary (etymology) Implicit in the name “algebraic geometry” is the relation between geometry and equations The qualitative study of systems of polynomial equations is the chief subject of commutative algebra as well But when we actually study a ring or a variety, we often have to know a great deal about it before understanding its equations Conversely, given a system of equations, it can be extremely difficult to analyze its qualitative properties, such as the geometry of the corresponding variety The theory of syzygies offers a microscope for looking at systems of equations, and helps to make their subtle properties visible This book is concerned with the qualitative geometric theory of syzygies It describes geometric properties of a projective variety that correspond to the numbers and degrees of its syzygies or to its having some structural property — such as being determinantal, or having a free resolution with some particularly simple structure It is intended as a second course in algebraic geometry and commutative algebra, such as I have taught at Brandeis University, the Institut Poincar´e in Paris, and the University of California at Berkeley What Are Syzygies? In algebraic geometry over a field K we study the geometry of varieties through properties of the polynomial ring S = K[x0 , , xr ] x Preface: Algebra and Geometry and its ideals It turns out that to study ideals effectively we we also need to study more general graded modules over S The simplest way to describe a module is by generators and relations We may think of a set A ⊂ M of generators for an S-module M as a map from a free S-module F = S A onto M , sending the basis element of F corresponding to a generator m ∈ A to the element m ∈ M Let M1 be the kernel of the map F → M ; it is called the module of syzygies of M corresponding to the given choice of generators, and a syzygy of M is an element of M — a linear relation, with coefficients in S, on the chosen generators When we give M by generators and relations, we are choosing generators for M and generators for the module of syzygies of M The use of “syzygy” in this context seems to go back to Sylvester [1853] The word entered the language of modern science in the seventeenth century, with the same astronomical meaning it had in ancient Greek: the conjunction or opposition of heavenly bodies Its literal derivation is a yoking together, just like “conjunction”, with which it is cognate If r = 0, so that we are working over the polynomial ring in one variable, the module of syzygies is itself a free module, since over a principal ideal domain every submodule of a free module is free But when r > it may be the case that any set of generators of the module of syzygies has relations To understand them, we proceed as before: we choose a generating set of syzygies and use them to define a map from a new free module, say F , onto M1 ; equivalently, we give a map φ1 : F1 → F whose image is M1 Continuing in this way we get a free resolution of M, that is, a sequence of maps ··· ✲ F2 ✲ F1 φ2 ✲ F φ1 ✲ M ✲ 0, where all the modules Fi are free and each map is a surjection onto the kernel of the following map The image Mi of φi is called the i-th module of syzygies of M In projective geometry we treat S as a graded ring by giving each variable x i degree 1, and we will be interested in the case where M is a finitely generated graded S-module In this case we can choose a minimal set of homogeneous generators for M (that is, one with as few elements as possible), and we choose the degrees of the generators of F so that the map F → M preserves degrees The syzygy module M1 is then a graded submodule of F , and Hilbert’s Basis Theorem tells us that M1 is again finitely generated, so we may repeat the procedure Hilbert’s Syzygy Theorem tells us that the modules M i are free as soon as i ≥ r The free resolution of M appears to depend strongly on our initial choice of generators for M, as well as the subsequent choices of generators of M , and so on But if M is a finitely generated graded module and we choose a minimal set of generators for M , then M1 is, up to isomorphism, independent of the minimal set of generators chosen It follows that if we choose minimal sets of generators at each stage in the construction of a free resolution we get a minimal free resolution of M that is, up to isomorphism, independent of all the choices made Since, by the Hilbert Syzygy Theorem, M i is free for i > r, we see that in the minimal free resolution Fi = for i > r + In this sense the minimal free resolution is finite: it has length at most r + Moreover, any free resolution of M can be derived from the minimal one in a simple way (see Section 1B) Preface: Algebra and Geometry xi The Geometric Content of Syzygies The minimal free resolution of a module M is a good tool for extracting information about M For example, Hilbert’s motivation for his results just quoted was to devise a simple formula for the dimension of the d-th graded component of M as a function of d He showed that the function d → dimK Md , now called the Hilbert function of M, agrees for large d with a polynomial function of d The coefficients of this polynomial are among the most important invariants of the module If X ⊂ P r is a curve, the Hilbert polynomial of the homogeneous coordinate ring SX of X is (deg X) d + (1 − genus X), whose coefficients deg X and 1−genus X give a topological classification of the embedded curve Hilbert originally studied free resolutions because their discrete invariants, the graded Betti numbers, determine the Hilbert function (see Chapter 1) But the graded Betti numbers contain more information than the Hilbert function A typical example is the case of seven points in P , described in Section 2C: every set of points in P in linearly general position has the same Hilbert function, but the graded Betti numbers of the ideal of the points tell us whether the points lie on a rational normal curve Most of this book is concerned with examples one dimension higher: we study the graded Betti numbers of the ideals of a projective curve, and relate them to the geometric properties of the curve To take just one example from those we will explore, Green’s Conjecture (still open) says that the graded Betti numbers of the ideal of a canonically embedded curve tell us the curve’s Clifford index (most of the time this index is less than the minimal degree of a map from the curve to P ) This circle of ideas is described in Chapter Some work has been done on syzygies of higher-dimensional varieties too, though this subject is less well-developed Syzygies are important in the study of embeddings of abelian varieties, and thus in the study of moduli of abelian varieties (for example [Gross and Popescu 2001]) They currently play a part in the study of surfaces of low codimension (for example [Decker and Schreyer 2000]), and other questions about surfaces (for example [Gallego and Purnaprajna 1999]) They have also been used in the study of Calabi–Yau varieties (for example [Gallego and Purnaprajna 1998]) What Does Solving Linear Equations Mean? A free resolution may be thought of as the result of fully solving a system of linear equations with polynomial coefficients To set the stage, consider a system of linear equations AX = 0, where A is a p × q matrix of elements of K, which we may think of as a linear transformation A F1 = K q ✲ K p = F Suppose we find some solution vectors X1 , , Xn These vectors constitute a complete solution to the equations if every solution vector can be expressed as a linear combination of them Elementary linear algebra shows that there are complete solutions consisting of xii Preface: Algebra and Geometry q − rank A independent vectors Moreover, there is a powerful test for completeness: A given set of solutions {Xi } is complete if and only if it contains q − rank A independent vectors A set of solutions can be interpreted as the columns of a matrix X defining a map X : F2 → F1 such that X A F2 ✲ F1 ✲ F0 is a complex The test for completeness says that this complex is exact if and only if rank A + rank X = rank F1 If the solutions are linearly independent as well as forming a complete system, we get an exact sequence ✲ F1 ✲ F0 X → F2 A Suppose now that the elements of A vary as polynomial functions of some parameters x0 , , xr , and we need to find solution vectors whose entries also vary as polynomial functions Given a set X1 , , Xn of vectors of polynomials that are solutions to the equations AX = 0, we ask whether every solution can be written as a linear combination of the Xi with polynomial coefficients If so we say that the set of solutions is complete The solutions are once again elements of the kernel of the map A : F = S q → F0 = S p , and a complete set of solutions is a set of generators of the kernel Thus Hilbert’s Basis Theorem implies that there exist finite complete sets of solutions However, it might be that every complete set of solutions is linearly dependent: the syzygy module M = ker A is not free Thus to understand the solutions we must compute the dependency relations on them, and then the dependency relations on these This is precisely a free resolution of the cokernel of A When we think of solving a system of linear equations, we should think of the whole free resolution One reward for this point of view is a criterion analogous to the rank criterion given above for the completeness of a set of solutions We know no simple criterion for the completeness of a given set of solutions to a system of linear equations over S, that is, for the exactness of a complex of free S-modules F2 → F1 → F0 However, if we consider a whole free resolution, the situation is better: a complex ✲ Fm ✲ ··· φm ✲ F1 φ2 ✲ F0 φ1 of matrices of polynomial functions is exact if and only if the ranks r i of the φi satisfy the conditions ri + ri−1 = rank Fi , as in the case where S is a field, and the set of points {p ∈ K r+1 | the evaluated matrix φi |x=p has rank < ri } has codimension ≥ i for each i (See Theorem 3.4.) This criterion, from joint work with David Buchsbaum, was my first real result about free resolutions I’ve been hooked ever since Experiment and Computation A qualitative understanding of equations makes algebraic geometry more accessible to experiment: when it is possible to test geometric properties using their equations, it becomes possible to make constructions and decide their structure by computer Sometimes Preface: Algebra and Geometry xiii unexpected patterns and regularities emerge and lead to surprising conjectures The experimental method is a useful addition to the method of guessing new theorems by extrapolating from old ones I personally owe to experiment some of the theorems of which I’m proudest Number theory provides a good example of how this principle can operate: experiment is much easier in number theory than in algebraic geometry, and this is one of the reasons that number theory is so richly endowed with marvelous and difficult conjectures The conjectures discovered by experiment can be trivial or very difficult; they usually come with no pedigree suggesting methods for proof As in physics, chemistry or biology, there is art involved in inventing feasible experiments that have useful answers A good example where experiments with syzygies were useful in algebraic geometry is the study of surfaces of low degree in projective 4-space, as in work of Aure, Decker, Hulek, Popescu and Ranestad [Aure et al 1997] Another is the work on Fano manifolds such as that of of Schreyer [2001], or the applications surveyed in [Decker and Schreyer 2001, Decker and Eisenbud 2002] The idea, roughly, is to deduce the form of the equations from the geometric properties that the varieties are supposed to possess, guess at sets of equations with this structure, and then prove that the guessed equations represent actual varieties Syzygies were also crucial in my work with Joe Harris on algebraic curves Many further examples of this sort could be given within algebraic geometry, and there are still more examples in commutative algebra and other related areas, such as those described in the Macaulay Book [Decker and Eisenbud 2002] Computation in algebraic geometry is itself an interesting field of study, not covered in this book It has developed a great deal in recent years, and there are now at least three powerful programs devoted to computation in commutative algebra, algebraic geometry and singularities that are freely available: CoCoA, Macaulay 2, and Singular Despite these advances, it will always be easy to give sets of equations that render our best algorithms and biggest machines useless, so the qualitative theory remains essential A useful adjunct to this book would be a study of the construction of Gră obner bases which underlies these tools, perhaps from [Eisenbud 1995, Chapter 15], and the use of one of these computing platforms The books [Greuel and Pfister 2002, Kreuzer and Robbiano 2000] and, for projective geometry, the forthcoming book [Decker and Schreyer ≥ 2004], will be very helpful What’s In This Book? The first chapter of this book is introductory: it explains the ideas of Hilbert that give the definitive link between syzygies and the Hilbert function This is the origin of the modern theory of syzygies This chapter also introduces the basic discrete invariants of resolution, the graded Betti numbers, and the convenient Betti diagrams for displaying them At this stage we still have no tools for showing that a given complex is a resolution, and in Chapter we remedy this lack with a simple but very effective idea of Bayer, Peeva, and Sturmfels for describing some resolutions in terms of labeled simplicial complexes With These software packages are freely available for many platforms, at the websites cocoa.dima.unige.it, www.math.uiuc.edu/Macaulay2 and www.singular.uni-kl.de, respectively These web sites are good sources of further information and references xiv Preface: Algebra and Geometry this tool we prove the Hilbert Syzygy Theorem and we also introduce Koszul homology We then spend some time on the example of seven points in P , where we see a deep connection between syzygies and an important invariant of the positions of the seven points In the next chapter we explore a case where we can say a great deal: sets of points in P Here we characterize all possible resolutions and derive some invariants of point sets from the structure of syzygies The following Chapter introduces a basic invariant of the resolution, coarser than the graded Betti numbers: the Castelnuovo–Mumford regularity This is a topic of central importance for the rest of the book, and a very active one for research The goal of Chapter 4, however, is modest: we show that in the setting of sets of points in P r the Castelnuovo–Mumford regularity is the degree needed to interpolate any function as a polynomial function We also explore different characterizations of regularity, in terms of local or Zariski cohomology, and use them to prove some basic results used later Chapter is devoted to the most important result on Castelnuovo–Mumford regularity to date: the theorem by Castelnuovo, Mattuck, Mumford, Gruson, Lazarsfeld, and Peskine bounding the regularity of projective curves The techniques introduced here reappear many times later in the book The next chapter returns to examples We develop enough material about linear series to explain the free resolutions of all the curves of genus and in complete embeddings This material can be generalized to deal with nice embeddings of any hyperelliptic curve Chapter is again devoted to a major result: Green’s Linear Syzygy theorem The proof involves us with exterior algebra constructions that can be organized around the Bernstein–Gelfand–Gelfand correspondence, and we spend a section at the end of Chapter exploring this tool Chapter is in many ways the culmination of the book In it we describe (and in most cases prove) the results that are the current state of knowledge of the syzygies of the ideal of a curve embedded by a complete linear series of high degree — that is, degree greater than twice the genus of the curve Many new techniques are needed, and many old ones resurface from earlier in the book The results directly generalize the picture, worked out much more explicitly, of the embeddings of curves of genus and We also present the conjectures of Green and Green–Lazarsfeld extending what we can prove No book on syzygies written at this time could omit a description of Green’s conjecture, which has been a wellspring of ideas and motivation for the whole area This is treated in Chapter However, in another sense the time is the worst possible for writing about the conjecture, since major new results, recently proven, are still unpublished These results will leave the state of the problem greatly advanced but still far from complete It’s clear that another book will have to be written some day Finally, I have included two appendices to help the reader: Appendix explains local cohomology and its relation to sheaf cohomology, and Appendix surveys, without proofs, the relevant commutative algebra I can perhaps claim (for the moment) to have written the longest exposition of commutative algebra in [Eisenbud 1995]; with this second appendix I claim also to have written the shortest! 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point of an abstract algebraic variety”, Trans Amer Math Soc 62 (1947), 1–52 This is page 224 Printer: Opaque this Index Abel–Jacobi map, 140, 141 acyclic complex, 15 adjunction formula, 169, 172 Akin, Kaan, 209 almost regular, 56, 66 Altman, Allen, 135 ample, 89, 135, see also very ample Arbarello, Enrico, 140, 147, 153, 168 arithmetically Cohen–Macaulay, see under Cohen–Macaulay Artin–Rees Lemma, 177, 186 Artinian module, 54, 146 Artinian ring, 132, 203 associated prime, 191 astronomy, x Aure, Alf, xiii Auslander, Maurice, 10, 204 Auslander–Buchsbaum formula, 10, 12, 21, 24, 25, 30, 33, 54, 84, 85, 101, 195 Auslander–Buchsbaum–Serre characterization, 11, 196 Bertini’s Theorem, 68 Betti diagram, 7, 29, 37, 38, 42, 84, 94, 129 of linear strand, 110 Betti numbers, 8, see also graded Betti numbers and Hilbert function, bounds on, 18 graded, xi, xiii B´ ezout’s Theorem, 37, 95 blowup, 105, 200 Bonnin, Freddy, xv Boutot, Jean-Fran¸cois, 203 Brandeis University, ix Bridgeland, Tom, 127 Brill–Noether Theorem, 152–154, 167 Brodmann, Markus P., 175 Bruns, Winfried, 113, 194 Buchsbaum surface, 68 Buchsbaum, David, xv, 10, 31, 102, 204, see also Akin, Kaan; Auslander–Buchsbaum Buchsbaum–Eisenbud Theorem, 31 Burch, Lindsay, 30, 35 Babson, Eric, xv basepoint of linear series, 87 Bayer, David, xiii, xv, 13, 16, 62, 68 Be˘ılinson, A A., 127 Bernstein–Gelfand–Gelfand correspondence, xiv, xv, 109, 125, 127, 204 Calabi–Yau varieties, xi canonical divisor, 135 canonical model, 168–173 canonical module, 113 canonical sheaf, 135 References Cartan complex (resolution), 117, 118, 127, 130, 131 Cartan, Henri, 204 Castelnuovo theory, 29 Castelnuovo, Guido, xiv, 52, 65, 67, 136, 168 Castelnuovo–Mumford regularity, xiv, 9, 21, 51–66, 128, 136, 170 and cohomology, 54 and interpolation problem, 59 catalecticant, 83 Caubel, Cl´ ement, xv Caviglia, Giulio, 62 Cayley, Arthur, 2, 83 ˇ Cech cohomology, 176, 177, 180–183, 186 ceiling, 144 chain of ideals, 193 characteristic p, 21, 180 three, 97 two, 97 vector, 103 zero, 2, 21, 41, 42, 47, 68, 103, 104, 120, 140, 161, 167, 170, 200, 209 Chen, Zhihua, 168 Ciliberto, Ciro, 36 Clifford index, xi, 165–173 Clifford’s Theorem, 107, 166, 173 CoCoA, xiii Cohen’s Structure Theorem, 197 Cohen, Irvin S., 200 Cohen–Macaulay, 159 arithmetically, 67, 68, 161, 204 domain, 85 module, 53, 61, 64, 90, 95, 96, 101 property, 136, 166, 167, 169–173 ring, 32, 36, 84, 136, 143, 185, 195, 200–204, 208, 209, 211 coherent sheaf, 62, 65, 71, 72, 75, 80, 127, 128, 131, 142, 175, 186, 202 Tate resolution of, 129 cohomological degree, 131, 177, 205 cohomology, 54 cohomology, local, 175–187 column, generalized, 120 complete intersection, 11, 36, 37, 39, 95, 169, 171, 180, 194, 203 linear series, 86–88, 90–92 complex, see also Cartan linear strand of a, see under linear minimal, regularity of, 51 simplicial, see under simplicial 225 trivial, Conca, Aldo, 103 cone, see also mapping cone over a set, 67 over conic, 98 over Veronese surface, 212 quadric, 200 conic, 82 points on a, 37 series, 87, 88 standard, 82, 98 connected in codimension 1, 68 coordinate ring, Cornalba, Maurizio, see Arbarello, Enrico cotangent space, 195 Cramer’s Rule, 210 Criterion of Exactness, 31 cubic, 94, 97, 110, 169, 172 standard twisted, 129 curve, see also under elliptic, hyperelliptic, etc of high degree, 135 cyclic module, 19 Decker, Wolfram, xi, xiii, 131 degree cohomological, 131, 177, 205 matrix, 35, 36 δ-gonal, see gonal depth, 30, 53, 56 infinity, 31 of ideal, 194 Derksen, Harm, 66 determinantal ideal, 30, 31, 207–209 Dictionary Theorem, 116, 118, 119 differential, 5–8, 14–19, 73, 93, 98, 111, 114–117, 127, 148, 150, 152 graded module, 128 dimension of linear series, 86 of ring, 193 discrete valuation ring, 196 discriminant, divided power, 120 divisor class, 99 duality, 206, see also self-duality, Serre duality functor, 125, 126 Koszul, 204 local, 54, 65, 170, 181, 182, 195 Eagon, John A., 207, 208 Eagon–Northcott complex, 69, 74, 84, 85, 90, 92–95, 101–103, 112, 129, 159, 209–213 Edmonds’ Theorem, 26 Eilenberg, Samuel, 204 226 References Eisenbud, David, 10, 31, 68, 102, 109, 113, 115, 123, 125, 131, 140, 145, 153, 156, 167, 170, 172, 193 Eisenbud–Goto conjecture, 68 elementary divisor, 208 elliptic curve, 78, 94, 96 normal, 94–97, 101, 102, 107 embedded component, 192 embedded prime, 191 embedding complete, xiv of high degree, 135 Euler characteristic, 184 Evans, E Graham, 185 experiment as a source of conjectures, xiii exterior algebra, 71–72, 76, 77, 80, 109–133, 205–206, 209, 212 exterior minor, 120–124 face of a complex, 13 facet, 13 factorial domain, 196, 199 Fano manifolds, xiii fibered algebraic set, 85 filtration, 76–78 finite morphism, 200 finite sets of points, 29, 52, 55 Fitting ideal, 70, 74, 207–208 Fitting’s Lemma, 109, 119 exterior, 123 Fløystad, Gunnar, 109, 115, 125, 131 free, see under resolution, module Fu, Baohua, xv full subcomplex, 16 Fulton, William, 154 Gaeta set, 47 Gaeta, Federigo, 46 Gale transform, 29 Gallego, Francisco Javier, xi Gelfand, see also Bernstein–Gelfand–Gelfand Gelfand, I M., 204 general position linearly, 20 generalized local ring, 190 generalized row, column, entry, 84, 120 1-generic matrices, 81 1-generic matrix, 84, 90, 95, 98, 105 generic set of points, 47 genus 0, 81, 135 genus 1, 81, 94, 95, 135 geometric realization, 14 Geramita, Anthony V., 36, 41, 48, 104 Giaimo, Daniel, 68 Gieseker, D., 153 global section, 63, 76, 87, 96, 100, 139–142, 147, 154 nonnegatively twisted, 113 twisted, 52, 132 Gold, Leah, xv gonal, 159, 167 gonality, 159, 166 Gorenstein property, 126 Gorenstein ring, 170 Gorenstein subschemes, 135 Goto, Shiro, 68, 190 Gră obner basis, 48, 62, 106, 130 grade of an ideal, 30 grade of ideal, 194 graded Betti number, 18, 20, 22, 64, 137, 143–145, 157, 171 Betti numbers, 90 free complex, 14 free resolution, 13, 29, 34, 35, 53, 114 modules, equivalence to linear free complexes, 115, 127 Greek, ix, x Green’s Conjecture, xi, xiv, 20, 168–172 statement, 170 Green’s Theorem, see Linear Syzygy Theorem Green, Mark L., xv, 70, 109, 113, 115, 136, 158, 159, 168, 170, 171 Green–Lazarsfeld Conjecture, 171 Gregory, David A, 41, 48 Griffith, Phillip, 185 Griffiths, Phillip A., 105, 168, see also Arbarello, Enrico Gross, Mark, xi Grothendieck’s Vanishing Theorem, 159, 185 Grothendieck, Alexandre, 175, 177 Gruson, Laurent, xiv, 62, 67, 103 Gruson–Lazarsfeld–Peskine Theorem, 67, 83, 103, 129 Hankel matrix, 83 Harris, Joe, xiii, xv, 105, 140, 153, 161, 168, 172, see also Arbarello, Enrico Hartshorne, Robin, 50, 131, 140, 162, 175, 177, 180 Herzog, Jă urgen, 36, 113, 194 high degree, embedding of, 135 Hilbert Basis Theorem, x, xii, 2, Hilbert function, xi, xiii, 1, 3, 4, 10, 11, 20, 29, 36, 47, 53, 64, 136 and Betti numbers, Hilbert polynomial, xi, 1, 4, 11, 20, 29, 53, 58, 90 Hilbert series, 11 References Hilbert Syzygy Theorem, x, xiv, 13, 53 proof, 18 statement, Hilbert, David, 2–3, 10, 30 Hilbert–Burch Theorem, 30, 34, 35, 111 Hironaka, Heisuke, 200 Hirschowitz, Andr´ e, 172 Hochster, Melvin, 193, 203, 208 homogeneous algebra, 190 homology, reduced, 15 Hulek, Klaus, xiii Huneke, Craig, 145, 170 hyperelliptic curve, 81, 103, 148, 157–160, 166 hyperplane series, 87 hypersurface rings, 11 ideal membership, 48 of exterior minors, 120 reduced, 50 independent conditions, 59, 89 Institut Poincar´ e, ix, xv integers, ring of, 197 integral closure, 197–199 interpolation, 52, 59 intersection form, 98 invariants, 1, 10 of a resolution, 34 of a set of points, 34 irrelevant simplicial complex, 14 Kapranov, M M., see Gelfand, I M Kempf, George R., 93, 153 Kirkup, George, xv Kleiman, Steven L., 135, 153 Koh, Jee, 62, 113, 150, 168 Koh, Jee Heub, xv Koszul cohomology, 70, 135 Koszul complex, 3, 13, 17, 64, 65, 71, 72, 112, 115, 117, 150, 152, 204–207, 209, 210, 212 Koszul duality, 204 Koszul homology, xiv Kronecker, Leopold, 98 Krull dimension, 193, 196 Krull, Wolfgang, 195 Kwak, Sijong, 68 L’Hˆ opital, Guillaume Fran¸cois Antoine, Marquis de, labeled simplicial complex, 14 Laksov, Dan, 153 Lange, Herbert, 167 Laplace expansion, 23 Lasker, Emanuel, 190, 200, 201 227 Laurent polynomial, 10, 64 Lazarsfeld, Robert, xiv, 62, 67, 68, 136, 154, 158, 168, 171 Leibniz, Gottfried Wilhelm, Freiherr von, line bundles on curves, 78 linear form, 57, 60, 64, 66 free complex, 114 presentation, 70 projection, 88 series, 81, 86 strand, 109–113, 115, 116, 118–119, 124 second, 110, 111 syzygy, 23 Linear Syzygy Theorem, xiv, 109–113, 116, 131 proof, 116, 124 linearly general position, 20 linearly normal, 88 local cohomology, 175–187 and cohomology of sheaves, 183 and Zariski cohomology, 183 local ring, regular, 11, 196, 199 Lvovsky, S., 75 Lyubeznik, Gennady, 175, 180 Macaulay (software), xiii, 173 Macaulay, Francis S., 190, 200, 208 MacRae, R E., 10 Manin, Yuri I., 127, 204 mapping cone, 93–95, 101, 206, 210 mapping cylinder, 94 matrix factorization, 12 matrix of degrees of a presentation matrix, 36 Mattuck, Arthur, xiv, 136 Maugendre, H` el´ ene, xv maximal homogeneous ideal, 190 Miller, Ezra, 48 minimal complex, free resolution, x, 1, 84, 90, 92–94, 102, 106, 109, 110 construction, definition, graded, 29, 34, 35, 53, 114 regularity, 51 uniqueness, set of generators, x minimal component, 192 minimal prime, 191 minor (of matrix), 23, 31, 69, 70, 76, 80, 83–86, 91, 100, 104, see also exterior minor ideal of –s, 90 monodromy, 161–163 228 References monomial curve, 80 monomial ideal, 13, 16, 17, 40–42, 46, 48–50 primary decomposition, 49 square-free, 50 Mukai, Shigeru, 172 multiplication map, 96, 115, 118, 124 table, 83, 85, 97 Mumford, David, xiv, 52, 62, 65, 68, 136 Nagata, Masayoshi, 193 Nakayama’s Lemma, 6, 7, 55, 59, 71, 80, 125, 185, 190, 195–197 skew-commutative, 114 statement, node of a complex, 13 Noether’s Fundamental Theorem, 201 Noether, Emmy, 190, 200 Noether, Max, 168, 200, 201, see Brill–Noether Theorem nondegenerate curve, 82 irreducible of degree = dim, 90 of degree = dim, 89 ideal, 64 linear series, 86–88 variety, 64 nonzerodivisor, 30, 33, 41, 56, 59, 60 d-normal, 65 normal ring, 88, 197 normal variety, 88 normalization, 198 Northcott, Douglas Geoffrey, 30, 207 Nullstellensatz, 192 number theory, xiii Ogus, Arthur, 175 Orecchia, Ferruccio, 36 orientation, 14 Oxford English Dictionary, ix partition, 46 Peeva, Irena, xiii, xv, 13, 16, 68 Perkins, Pat, xv permanent, 120 Peskine, Christian, xiv, 62, 67, 103 Picard group, 98 Picard variety, 139–141, 147, 153, 158 polarization, 40, 50 Polishchuk, Alexander, 127 polynomial ring, 31 Popescu, Sorin, xi, xiii, xv, 170 Previato, Emma, xv primary decomposition, 49, 190, 192 primary ideal, 191 prime avoidance, 191 prime decomposition, 190 prime ideal, 191 Principal Ideal Theorem, 194, 195, 201, 207 generalized, 208 projective dimension, 194 of an ideal of points, 30 property Np , 135 Purnaprajna, B P., xi quadratic form, 23, 37, 39, 83, 87, 95, 170, 172 quadric, 200 quantum groups, 204 quartic curve, 110, 171, 172, 199, 204, 208 quasicoherent sheaf, 183 quintic curve, 167 radical, 192 Ramanan, Sundararaman, 172 Ranestad, Kristian, xiii, 104 rank of a map of free modules, 31 Rathmann, Jă urgen, 156, 161 rational curves, 78 rational function, 11, 96 rational map associated with linear series, 87 rational normal curve, 22, 83, 88–90, 92, 171 ideal of, 103 standard, 82 reduced ideal, 50 Rees algebra, 200 Rees, David, see Artin–Rees Lemma regular, see also under local ring element, 56 sequence, 95, 193, 195 regularity, see also Castelnuovo–Mumford regularity of a Cohen–Macaulay module, 53 of a module, 54 of a projective curve, 67 d-regularity, 56, 62, 65 representation theory, 127, 196 resolution of singularities, 198–200 resolution, free, x, 3, 29, 101 finite, 33 graded, minimal, see under minimal Reynolds operator, 203 Riemann, Georg F B., 200 Riemann–Roch Theorem, 89, 95, 137, 141, 143, 147, 152–154, 158, 160 Ringel, Claus Michael, 127 Roberts, Joel L., 203 References Roberts, Leslie, 41, 48 rolling, 162 Room, Thomas Gerald, 85, 103 Sard’s Theorem, 162 schemes, 29 Schenck, Hal, xv Schepler, Daniel, 175 Schmitt, Thomas, 180 Schreyer’s Theorem, 154 Schreyer, Frank-Olaf, xi, xiii, xv, 104, 109, 115, 125, 131, 153, 154, 167, 172 scroll, 209–213 rational normal, 98, 103, 210–212 n-secant, 79 second linear strand, 110 Segre embedding, 125 self-duality, 126, 132, 170, 206 Serre duality, 136, 140, 141, 143, 166 Serre’s Criterion, 198, 200 Serre’s Vanishing Theorem, 159, 184, 185 Serre, Jean-Pierre, 204, 207 Sharp, Rodney Y., 175 sheaf cohomology, xiv, 52, 54, 62, 65, 73, 100–101, see also coherent sheaf sheafification, 70–73, 100, 128, 129, 131 Sidman, Jessica, xv, 66 simplex, 13 simplicial complex, 13 irrelevant, 14 labeled, xiii, 14 void, 13 Singular (software), xiii skew lines, 25 skew-commutative, 114 Smith, Gregory, xv special line bundle, 166 square-free monomial, 14, 50, 114 standard graded algebra, 190 Stanley–Reisner correspondence, 14 Stillman, Michael, xv, 62, 150, 168 strand, see under linear, cubic strands of the resolution, 135 strictly commutative, 114 structure theorem, 11 Sturmfels, Bernd, xiii, xv, 13, 16, 48, 68 Stă uckrad, Jă urgen, 68, 180 Sylvester, James Joseph, x, symbolic power, 193 symmetric power, 84, 100–102, 112, 118, 121–125, 127, 130–133 syzygy astronomical, x 229 definition, ix, etymology, ix, x linear, 23 of monomial ideals, 16 origin of, theorem, see under Hilbert tangent space, 195 Tate resolution, 109, 126–132 tautological subbundle, 71, 72, 75 restricted, 135, 138 Taylor, Diana, 17 complex, 17 Teixidor I Bigas, Montserrat, 172 Thomas, Rekha, xv toric variety, 68 transcendence degree, 194 trigonal, 159 Trung, Ngˆ o Viˆ et, 36 Turner, Simon, xv twisted cubic, 22, 25 Ulrich, Bernd, 145 uniform position, 29 Uniform Position Principle, 161 University of California at Berkeley, ix, xv Unmixedness Theorem, 201 Veronese embedding, 159, 171, 203, 212 vertex of a complex, 13 position of, 14 very ample, 88, 94, 168, 169 Vogel, Wolfgang, 68, 180 void simplicial complex, 13 Voisin, Claire, 172 Waring’s problem, 104 Watanabe, Keiichi, 190 weak d-regularity, 56 Weierstrass normal form, 97 Weierstrass, Karl Wilhelm Theodor, 98 Weiss, Arthur, xv Weyman, Jerzy, xv, 109, 123, see also Akin, Kaan Yau, Qin, 168 Yuzvinsky, Sergey, xv Zariski (co)tangent space, 195 Zariski cohomology, 182, 183 Zariski, Oscar, 193, 195, 198 Zaslow, Eric, 127 Zelevinsky, A V., see Gelfand, I M ... each face of ∆ as the convex hull of its vertex points; the realization of ∆ is then the union of these faces An orientation of a simplicial complex consists of an ordering of the vertices of ∆ Thus... bases of F and G, the generator of I that is the image of the i-th basis vector of G is ±a times the determinant of the submatrix of M formed from all except the i-th row Moreover , the grade of. .. the free module of rank with generator in degree a The are thus the degrees of the minimal generators of I The degree of the (i, j) entry of the matrix M is then bj − As we shall soon see, the
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