DOCSLIB.ORG

Joachim Jelisiejew – Hilbert Schemes of Points And

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Joachim Jelisiejew – Hilbert Schemes of Points And University of Warsaw Faculty of Mathematics, Informatics and Mechanics Joachim Jelisiejew Hilbert schemes of points and their applications PhD dissertation Supervisor dr hab. Jarosław Buczyński Institute of Mathematics University of Warsaw and Institute of Mathematics Polish Academy of Sciences Auxiliary Supervisor dr Weronika Buczyńska Institute of Mathematics University of Warsaw Author’s declaration: I hereby declare that I have written this dissertation myself and all the contents of the dissertation have been obtained by legal means. May 9, 2017 ................................................... Joachim Jelisiejew Supervisors’ declaration: The dissertation is ready to be reviewed. May 9, 2017 ................................................... dr hab. Jarosław Buczyński May 9, 2017 ................................................... dr Weronika Buczyńska Abstract This thesis is concerned with deformation theory of finite subschemes of smooth varieties. Of central interest are the smoothable subschemes (i.e., limits of smooth subschemes). We prove that all Gorenstein subschemes of degree up to 13 are smoothable. This result has immediate applications to finding equations of secant varieties. We also give a description of nonsmoothable Gorenstein subschemes of degree 14, together with an explicit condition for smoothability. We prove that being smoothable is a local property, that it does not depend on the embedding and it is invariant under a base field extension. The above results are equivalently stated in terms of the Hilbert scheme of points, which is the moduli space for this deformation problem. We extensively use the combinatorial framework of Macaulay’s inverse systems. We enrich it with a pro-algebraic group action and use this to reprove and extend recent classification results by Elias and Rossi. We provide a relative version of this framework and use it to give a local description of the universal family over the Hilbert scheme of points. We shortly discuss history of Hilbert schemes of points and provide a list of open questions. Keywords: deformation theory, Hilbert scheme, Gorenstein algebra, inverse system, apolar- ity, smoothability, classification of finite commutative algebras. AMS MSC 2010 classification: 14C05, 14B07, 13N10, 13H10. Contents 1 Introduction 1 1.1 Overview and main results . .3 1.2 Application to secant varieties . .5 1.3 Application to constructing r-regular maps . .6 1.4 A brief historical survey . .8 1.5 Open problems . 12 I Finite algebras 14 2 Basic properties of finite algebras 15 2.1 Gorenstein algebras . 15 2.2 Hilbert function of a local algebra . 18 2.3 Hilbert function of a local Gorenstein algebra . 20 2.4 Betti tables, Boij and Söderberg theory . 23 3 Macaulay’s inverse systems (apolarity) 25 3.1 Definition of contraction action . 25 3.2 Automorphisms and derivations of the power series ring . 28 3.3 Classification of local embedded Gorenstein algebras via apolarity . 30 3.4 Hilbert functions of Gorenstein algebras in terms of inverse systems . 34 3.5 Standard forms of dual generators . 36 3.6 Simplifying dual generators . 38 3.7 Examples I — compressed algebras . 41 3.8 Examples II — (1; 3; 3; 3; 1) .............................. 44 3.9 Examples III — preliminaries for Chapter 6 . 46 II Hilbert schemes 48 4 Preliminaries 49 4.1 Moduli spaces of finite algebras, Hilbert schemes . 49 4.2 Base change of Hilbert schemes . 54 4.3 Loci of Hilbert schemes of points . 55 4.4 Relative Macaulay’s inverse systems . 60 4.5 Homogeneous forms and secant varieties . 66 5 Smoothings 70 5.1 Abstract smoothings . 70 5.2 Comparing abstract and embedded smoothings . 74 5.3 Embedded smoothability depends only on singularity type . 75 5.4 Comparing embedded smoothings and the Hilbert scheme . 76 5.5 Smoothings over rational curves . 78 5.6 Examples of nonsmoothable finite schemes . 79 5.7 Example of smoothings: one-dimensional torus limits . 82 III Applications 88 6 Gorenstein loci for small number of points 89 6.1 Ray families . 90 6.2 Ray families from Macaulay’s inverse systems. 93 6.3 Tangent preserving ray families . 96 6.4 Proof of Theorem 6.1 — preliminaries . 101 6.5 Proof of Theorem 6.1 — smoothability results . 105 6.6 Proof of Theorem 6.3 — the nonsmoothable component (1,6,6,1) . 109 7 Small punctual Hilbert schemes 116 Chapter 1 Introduction The Hilbert scheme of points on a smooth variety X is of central interest for several branches of mathematics: in commutative algebra, it is a moduli space of finite algebras (presented as quotients of a • fixed ring), in geometry and topology, it is a compact variety containing the space of tuples of points • on X; in many cases this space is dense, in algebraic geometry, its construction (1960-61) is one of the advances of the Grothendieck • school [Gro95], it found applications in constructing other moduli spaces and hyperkähler manifolds, and also in McKay correspondence and theory of higher secant varieties, in combinatorics, the Hilbert scheme appears in Haiman’s proofs of n! and Macdonald • positivity conjectures. Consider the set of finite algebras, presented as quotients of a fixed polynomial ring. There is a unique and natural topological space structure on this set, together with a sheaf of regular functions. These structures jointly give a scheme structure, called the Hilbert scheme of points on affine space, see Chapter 4.1 for precise definition. These structures are unique, but they are non-explicit and difficult to investigate; many open questions persist, despite continuous research, see Section 1.5. In this thesis we analyse the geometry of Hilbert schemes of points on smooth varieties, concentrating on the following question: What are the irreducible components of the Hilbert scheme of points? What are their intersections and singularities? An informal, intuitive view of geometry of the components is given on Figure 1 below. Our analysis of the Hilbert scheme as a moduli space of finite algebras requires tools for working with algebras themselves, which are developed in Part I. We then switch our attention to families of algebras (subschemes) in Part II and analyse Hilbert schemes for small numbers of points in Part III. Almost all of our original results presented in this thesis are also found in [Jel14, CJN15, Jel16, BJ17, Jel17, BJJM17]. 1 Figure 1.1: Bellis Hilbertis. Components of the Hilbert scheme of r points on a smooth variety X. Flower, petals and stem correspond to irreducible components of the Hilbert scheme of X. The analysis of their geometry is the main aim of this work. 1 The smoothable component, see Definition 4.23. It compactifies the space of r-tuples of points on X, thus has dimension r(dim X). For small r, it is the only component – there are no “petals” – see Section 5.6, Theorem 6.1 and also Problems 1.16, 1.17. 2 Points of the Hilbert scheme correspond to finite subschemes of X. General points on the smoothable component (“ladybirds”) correspond to tuples of points on X. Thus all subschemes corresponding to points on this component are limits of tuples of points; they are smoothable. 3 Every component intersects the smoothable one, see [Ree95]. Describing the intersection is subtle, see Problem 1.19 and Theorem 6.3 for an example involving cubic fourfolds. It is even hard to decide whether a given subscheme of X is smoothable, see Problem 1.18. 4 There is a single example [EV10], where components intersect away from the smoothable component. No singular points lying on a unique nonsmoothable component are known. 5 There are only few known components of dimension smaller than the smoothable one, see Section 5.6 and Problems 1.22, 1.23. 6 There are many examples of loci too large to fit inside the smoothable component, see Section 5.6, however the components containing these loci are not known. 7 Not much is known about the geometry and singularities of components other than the smoothable one, see Problem 1.13, Problem 1.14. 2 1.1 Overview and main results Part I gathers tools for studying finite local algebras over a field |, especially Gorenstein algebras. In this part, we speak the language of algebra; only basic background on commutative algebras and Lie theory is assumed. Part I is mostly prerequisite, although Sections 3.5-3.9 contain many original results published in [BJMR17, Jel17]. Theory of Macaulay’s inverse systems (also known as apolarity) is a central tool in our investigation. To an element f of a (divided power) polynomial ring P it assigns a finite local Gorenstein algebra Apolar (f), see Section 3.3. A pro-algebraic group G acts on P so that the orbits are isomorphism classes of Apolar ( ), see Section 3.3. In Section 3.6 we explicitly − describe the action of G and investigate its Lie group, building an effective tool to investigate isomorphism classes of algebras. We then give applications; as a sample result, we reprove (with weaker assumptions on the base field) the following main theorems of [ER12, ER15]. Theorem 1.1 (Example 3.35, Corollary 3.73). Let | be a field of characteristic = 2; 3. Let 6 n+1 (A; m; |) be a finite Gorenstein local |-algebra with Hilbert function (1; n; n; 1) or (1; n; 2 ; n; 1). Then A isomorphic to its associated graded algebra gr A. Interestingly, Theorem 1.1 fails for small characteristics, see Example 3.74. We also obtain genuine, down to earth classification results, such as the following. Proposition 1.2 (Example 3.75). Let | be an algebraically closed field of characteristic = 2; 3. 6 There are exactly eleven isomorphism types of finite local Gorenstein algebras with Hilbert func- tion (1; 3; 3; 3; 1), see Example 3.75 for their list. In Part II, our attention shifts towards families of algebras. Accordingly, we change the language from algebra to algebraic geometry, from finite algebras to finite schemes.
Load more

Recommended publications
  • Grothendieck Duality and Q-Gorenstein Morphisms 101
    GROTHENDIECK DUALITY AND Q-GORENSTEIN MORPHISMS YONGNAM LEE AND NOBORU NAKAYAMA Abstract. The notions of Q-Gorenstein scheme and of Q-Gorenstein mor- phism are introduced for locally Noetherian schemes by dualizing complexes and (relative) canonical sheaves. These cover all the previously known notions of Q-Gorenstein algebraic variety and of Q-Gorenstein deformation satisfying Koll´ar condition, over a field. By studies on relative S2-condition and base change properties, valuable results are proved for Q-Gorenstein morphisms, which include infinitesimal criterion, valuative criterion, Q-Gorenstein refine- ment, and so forth. Contents 1. Introduction 1 2. Serre’s Sk-condition 9 3. Relative S2-condition and flatness 24 4. Grothendieck duality 41 5. Relative canonical sheaves 63 6. Q-Gorenstein schemes 72 7. Q-Gorenstein morphisms 82 Appendix A. Some basic properties in scheme theory 105 References 108 1. Introduction arXiv:1612.01690v1 [math.AG] 6 Dec 2016 The notion of Q-Gorenstein variety is important for the minimal model theory of algebraic varieties in characteristic zero: A normal algebraic variety X defined over a field of any characteristic is said to be Q-Gorenstein if rKX is Cartier for some positive integer r, where KX stands for the canonical divisor of X. In some papers, X is additionally required to be Cohen–Macaulay. Reid used this notion essentially to define the canonical singularity in [48, Def. (1.1)], and he named the notion “Q- Gorenstein” in [49, (0.12.e)], where the Cohen–Macaulay condition is required. The notion without the Cohen–Macaulay condition appears in [24] for example.
    [Show full text]
  • Frobenius Categories, Gorenstein Algebras and Rational Surface Singularities
    FROBENIUS CATEGORIES, GORENSTEIN ALGEBRAS AND RATIONAL SURFACE SINGULARITIES OSAMU IYAMA, MARTIN KALCK, MICHAEL WEMYSS, AND DONG YANG Dedicated to Ragnar-Olaf Buchweitz on the occasion of his 60th birthday. Abstract. We give sufficient conditions for a Frobenius category to be equivalent to the category of Gorenstein projective modules over an Iwanaga–Gorenstein ring. We then apply this result to the Frobenius category of special Cohen–Macaulay modules over a rational surface singularity, where we show that the associated stable cate- gory is triangle equivalent to the singularity category of a certain discrepant partial resolution of the given rational singularity. In particular, this produces uncountably many Iwanaga–Gorenstein rings of finite GP type. We also apply our method to representation theory, obtaining Auslander–Solberg and Kong type results. Contents 1. Introduction 1 2. A Morita type theorem for Frobenius categories 3 2.1. Frobenius categories as categories of Gorenstein projective modules 3 2.2. Alternative Approach 6 2.3. A result of Auslander–Solberg 7 3. Frobenius structures on special Cohen–Macaulay modules 7 4. Relationship to partial resolutions of rational surface singularities 10 4.1. Gorenstein schemes and Iwanaga–Gorenstein rings 10 4.2. Tilting bundles on partial resolutions 11 4.3. Iwanaga–Gorenstein rings from surfaces 13 4.4. Construction of Iwanaga–Gorenstein rings 15 5. Relationship to relative singularity categories 18 6. Examples 20 6.1. Iwanaga–Gorenstein rings of finite GP type 20 6.2. Frobenius structures on module categories 22 6.3. Frobenius categories arising from preprojective algebras 23 References 24 1. Introduction This paper is motivated by the study of certain triangulated categories associated to rational surface singularities, first constructed in [IW2].
    [Show full text]
  • Non-Pfaffian Subcanonical Subschemes of Codimension 3
    ENRIQUES SURFACES AND OTHER NON-PFAFFIAN SUBCANONICAL SUBSCHEMES OF CODIMENSION 3 DAVID EISENBUD, SORIN POPESCU, AND CHARLES WALTER Abstract. We give examples of subcanonical subvarieties of codimen- sion 3 in projective n-space which are not Pfaffian, i.e. defined by the ideal sheaf of submaximal Pfaffians of an alternating map of vector bun- dles. This gives a negative answer to a question asked by Okonek [29]. Walter [36] had previously shown that a very large majority of sub- canonical subschemes of codimension 3 in Pn are Pfaffian, but he left open the question whether the exceptional non-Pfaffian cases actually occur. We give non-Pfaffian examples of the principal types allowed by his theorem, including (Enriques) surfaces in P5 in characteristic 2 and 7 a smooth 4-fold in PC. These examples are based on our previous work [14] showing that any strongly subcanonical subscheme of codimension 3 of a Noetherian scheme can be realized as a locus of degenerate intersection of a pair of Lagrangian (maximal isotropic) subbundles of a twisted orthogonal bundle. There are many relations between the vector bundles E on a nonsingular algebraic variety X and the subvarieties Z X. For example, a globalized form of the Hilbert-Burch theorem allows⊂ one to realize any codimension 2 locally Cohen-Macaulay subvariety as a degeneracy locus of a map of vector bundles. In addition the Serre construction gives a realization of any subcanonical codimension 2 subvariety Z X as the zero locus of a section of a rank 2 vector bundle on X. ⊂ The situation in codimension 3 is more complicated.
    [Show full text]
  • Frobenius Categories, Gorenstein Algebras and Rational Surface Singularities
    Frobenius categories, Gorenstein algebras and rational surface singularities Martin Kalck, Osamu Iyama, Michael Wemyss and Dong Yang Compositio Math. 151 (2015), 502{534. doi:10.1112/S0010437X14007647 Downloaded from https://www.cambridge.org/core. IP address: 170.106.33.42, on 28 Sep 2021 at 14:06:22, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1112/S0010437X14007647 Compositio Math. 151 (2015) 502{534 doi:10.1112/S0010437X14007647 Frobenius categories, Gorenstein algebras and rational surface singularities Martin Kalck, Osamu Iyama, Michael Wemyss and Dong Yang Dedicated to Ragnar-Olaf Buchweitz on the occasion of his 60th birthday Abstract We give sufficient conditions for a Frobenius category to be equivalent to the category of Gorenstein projective modules over an Iwanaga{Gorenstein ring. We then apply this result to the Frobenius category of special Cohen{Macaulay modules over a rational surface singularity, where we show that the associated stable category is triangle equivalent to the singularity category of a certain discrepant partial resolution of the given rational singularity. In particular, this produces uncountably many Iwanaga{ Gorenstein rings of finite Gorenstein projective type. We also apply our method to representation theory, obtaining Auslander{Solberg and Kong type results. Contents 1 Introduction 502 2 A Morita type theorem for Frobenius categories 505 3 Frobenius structures on special Cohen{Macaulay modules 511 4 Relationship to partial resolutions of rational surface singularities 513 5 Relationship to relative singularity categories 523 6 Examples 526 Acknowledgements 532 References 532 1. Introduction This paper is motivated by the study of certain triangulated categories associated to rational surface singularities, first constructed in [IW11].
    [Show full text]
  • Hypersurfaces with Nonsingular Normalization and Their Double Loci
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF ALGEBRA 53, 253-267 (1978) Hypersurfaces with Nonsingular Normalization and Their Double Loci JOEL ROBERTS* University of Minnesota, Minneapolis, Minnesota 55455 Communicated by D. Buchsbaum ReceivedAugust 22, 1977 INTRODUCTION Let X be an (irreducible) nonsingular projective variety, defined over the algebraically closed field K, and let V: X + X’ be a finite birational morphism, where X’ is a hypersurface in P+l, and Y = dim(X). (Such a morphism exists for any given X, by the results of [15, Sect. 21.) Thus, X is the normalization of X’. Moreover, there is an exact sequence of sheaves on x’: Let % be the conductor of rr,0z in Lo,, . By definition, V is the largest sheaf of ideals in 0,~ which annihilates (rr,O,)/U+ . We define the double locus of X’ to be the closed subscheme D C X’ whose structure sheaf is 0,*/V. Thus, x E D if and only if O,,,, is not normal. Since x’ has a nonsingular normalization, x E D if and only if x is a singular point of X’. In the first part of this paper (Sections 1 and 2 and Theorem 3.1) we prove that D is a Cohen-Macaulay scheme of pure dimension r - 1 and that there is a natural isomorphism wD z ((~,Ox)/~x,)(d - r - 2) where wg is the dualizing sheaf on D (defined as in [l]), and d is the degree of the hypenurface x’ C P r+l.
    [Show full text]
  • Various Differents for 0-Dimensional Schemes and Applications
    Dissertation Various Differents for 0-Dimensional Schemes and Applications Le Ngoc Long Eingereicht an der Fakult¨atf¨urInformatik und Mathematik der Universit¨atPassau als Dissertation zur Erlangung des Grades eines Doktors der Naturwissenschaften Submitted to the Department of Informatics and Mathematics of the Universit¨atPassau in Partial Fulfilment of the Requirements for the Degree of a Doctor in the Domain of Science Betreuer / Advisor: Prof. Dr. Martin Kreuzer Universit¨atPassau June 2015 Various Differents for 0-Dimensional Schemes and Applications Le Ngoc Long Erstgutachter: Prof. Dr. Martin Kreuzer Zweitgutachter: Prof. Dr. J¨urgenHerzog M¨undliche Pr¨ufer: Prof. Dr. Tobias Kaiser Prof. Dr. Brigitte Forster-Heinlein Der Fakult¨atf¨urInformatik und Mathematik der Universit¨atPassau vorgelegt im Juni 2015 To my parents, my wife and my daughter Acknowledgements This thesis would not have been written without the help and support of many people and institutions. First and foremost, I am deeply indebted to my supervisor, Prof. Dr. Martin Kreuzer, for his help, guidance and encouragement during my research. I thank him for also being free with his time to answer my many questions and for his endless support inside and outside mathematics. Many arguments in this thesis are extracted from his suggestions. I would especially like to express my gratitude to him for giving me and my wife an opportunity to study together in Passau. I would like to thank the hospitality and support of the Department of Mathematics and Informatics of the University of Passau during these years, and in particular, I am grateful to the members of my examination committee, especially Prof.
    [Show full text]
  • A Study of Quasi-Gorenstein Rings Ii: Deformation of Quasi-Gorenstein Property 3
    A STUDY OF QUASI-GORENSTEIN RINGS II: DEFORMATION OF QUASI-GORENSTEIN PROPERTY KAZUMA SHIMOMOTO, NAOKI TANIGUCHI, AND EHSAN TAVANFAR Abstract. In the present article, we investigate the following deformation problem. Let (R, m) be a local (graded local) Noetherian ring with a (homogeneous) regular element y ∈ m and assume that R/yR is quasi-Gorenstein. Then is R quasi-Gorenstein? We give positive answers to this problem under various assumptions, while we present a counter-example in general. We emphasize that absence of the Cohen-Macaulay condition requires delicate and subtle studies. Contents 1. Introduction 1 2. Notation and auxiliary lemmas 3 3. Deformation of quasi-Gorensteinness 4 4. Failure of deformation of quasi-Gorensteinness 11 5. Construction of quasi-Gorenstein rings which are not Cohen-Macaulay 13 References 16 1. Introduction In this article, we study the deformation problem of the quasi-Gorenstein property on local Noe- therian rings and construct some examples of non-Cohen-Macaulay, quasi-Gorenstein and normal domains. Recall that a local ring (R, m) is quasi-Gorenstein, if it has a canonical module ωR such that ωR =∼ R. For completeness, we state the general deformation problem as follows: Problem 1. Let (R, m) be a local (graded local) Noetherian ring and M be a nonzero finitely generated R-module with a (homogeneous) M-regular element y ∈ m. Assume that M/yM has P. Then does M possess P? arXiv:1810.03181v3 [math.AC] 8 Jun 2020 By specializing P=quasi-Gorenstein, we prove the following result by constructing an explicit example using Macaulay2 (see Theorem 4.2): Main Theorem 1.
    [Show full text]
  • Numerically Trivial Dualizing Sheaves
    Numerically Trivial Dualizing Sheaves Inaugural-Dissertation zur Erlangung des Doktorgrades der Mathematisch-Naturwissenschaftlichen Fakult¨at der Heinrich-Heine-Universit¨atD¨usseldorf vorgelegt von Andr´eSchell aus Duisburg D¨usseldorf, Dezember 2019 aus dem Mathematischen Institut der Heinrich-Heine-Universit¨atD¨usseldorf Gedruckt mit der Genehmigung der Mathematisch-Naturwissenschaftlichen Fakult¨atder Heinrich-Heine-Universit¨atD¨usseldorf Referent: Prof. Dr. Stefan Schr¨oer Korreferent: Juniorprof. Dr. Marcus Zibrowius Tag der m¨undlichenPr¨ufung:30.01.2020 Summary Motivation for this work was the question to what extent the Beauville{Bogomolov decom- position, or parts and variants of it, holds true in positive characteristic. The statement itself applies to compact K¨ahlermanifolds and its proof uses differential geometry. It can be transferred to smooth projective schemes X over the complex numbers by the Riemann existence theorem. Under the assumption that the dualizing sheaf !X is numerically triv- ial, it states that X admits a finite ´etalecovering X0 ! X such that the total space 0 X satisfies !X0 'OX0 and decomposes in a special way as a product, uniquely up to permutation. During the first, preparatory part of this thesis, the notion of numerical triviality is treated and characterized. The Albanese morphism serves as a central technical tool in what follows, so its core properties are developed in a self-contained way, also verifying its existence under more general assumptions than what seems to be recorded in literature. The second, main part begins to answer the question whether !X 2 Pic(X) has finite order as a consequence of its numerical triviality. More generally, this question can be asked if !X 'OX (KX ) is not necessarily invertible, as long as the canonical divisor KX is a Weil divisor.
    [Show full text]
  • The Projective Grometry of the Gale Transform
    Journal of Algebra 230, 127᎐173Ž. 2000 doi:10.1006rjabr.1999.7940, available online at http:rrwww.idealibrary.com on The Projective Geometry of the Gale Transform David Eisenbud1 Department of Mathematics, Uni¨ersity of California, Berkeley, California 94720 E-mail: [email protected] and Sorin Popescu Department of Mathematics, Columbia Uni¨ersity, New York, New York 10027 E-mail: [email protected] Communicated by Craig Huneke Received July 23, 1998 The Gale transform, an involution on sets of points in projective space, appears in a multitude of guises and in subjects as diverse as optimization, coding theory, theta functions, and recently in our proof that certain general sets of points fail to satisfy the minimal free resolution conjectureŽ see Eisenbud and Popescu, 1999, In¨ent. Math. 136, 419᎐449. In this paper we reexamine the Gale transform in the light of modern algebraic geometry. We give a more general definition in the context of finiteŽ. locally Gorenstein subschemes. We put in modern form a number of the more remarkable examples discovered in the past, and we add new constructions and connections to other areas of algebraic geometry. We generalize Goppa's theorem in coding theory and we give new applications to Castelnuovo theory. We also give references to classical and modern sources. ᮊ 2000 Academic Press CONTENTS I. The Gale transform 1. History 2. The scheme-theoretic definition 3. A generalized Goppa theorem 1 Both authors are grateful to the NSF for support during the preparation of this work. It was in joint work with David Buchsbaum that the first author first became familiar with the notion of a Gorenstein ring.
    [Show full text]
  • Deformation Theory
    Math 274 Lectures on Deformation Theory Robin Hartshorne c 2004 Preface My goal in these notes is to give an introduction to deformation theory by doing some basic constructions in careful detail in their sim- plest cases, by explaining why people do things the way they do, with examples, and then giving some typical interesting applications. The early sections of these notes are based on a course I gave in the Fall of 1979. Warning: The present state of these notes is rough. The notation and numbering systems are not consistent (though I hope they are consis- tent within each separate section). The cross-references and references to the literature are largely missing. Assumptions may vary from one section to another. The safest way to read these notes would be as a loosely connected series of short essays on deformation theory. The order of the sections is somewhat arbitrary, because the material does not naturally fall into any linear order. I will appreciate comments, suggestions, with particular reference to where I may have fallen into error, or where the text is confusing or misleading. Berkeley, September 6, 2004 i ii CONTENTS iii Contents Preface . i Chapter 1. Getting Started . 1 1. Introduction . 1 2. Structures over the dual numbers . 4 3. The T i functors . 11 4. The infinitesimal lifting property . 18 5. Deformation of rings . 25 Chapter 2. Higher Order Deformations . 33 6. Higher order deformations and obstruction theory . 33 7. Obstruction theory for a local ring . 39 8. Cohen–Macaulay in codimension two . 43 9. Complete intersections and Gorenstein in codimension three .
    [Show full text]
  • L'institut Fourier
    R AN IE N R A U L E O S F D T E U L T I ’ I T N S ANNALES DE L’INSTITUT FOURIER Robin HARTSHORNE, Irene SABADINI & Enrico SCHLESINGER Codimension 3 Arithmetically Gorenstein Subschemes of projective N-space Tome 58, no 6 (2008), p. 2037-2073. <http://aif.cedram.org/item?id=AIF_2008__58_6_2037_0> © Association des Annales de l’institut Fourier, 2008, tous droits réservés. L’accès aux articles de la revue « Annales de l’institut Fourier » (http://aif.cedram.org/), implique l’accord avec les conditions générales d’utilisation (http://aif.cedram.org/legal/). Toute re- production en tout ou partie cet article sous quelque forme que ce soit pour tout usage autre que l’utilisation à fin strictement per- sonnelle du copiste est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.cedram.org/ Ann. Inst. Fourier, Grenoble 58, 6 (2008) 2037-2073 CODIMENSION 3 ARITHMETICALLY GORENSTEIN SUBSCHEMES OF PROJECTIVE N-SPACE by Robin HARTSHORNE, Irene SABADINI & Enrico SCHLESINGER (*) Abstract. — We study the lowest dimensional open case of the question N whether every arithmetically Cohen–Macaulay subscheme of P is glicci, that is, 3 whether every zero-scheme in P is glicci. We show that a general set of n > 56 3 points in P admits no strictly descending Gorenstein liaison or biliaison. In order to prove this theorem, we establish a number of important results about arithmeti- 3 cally Gorenstein zero-schemes in P .
    [Show full text]
  • GOOD IDEALS in GORENSTEIN LOCAL RINGS 1. Introduction In
    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 353, Number 6, Pages 2309{2346 S 0002-9947(00)02694-5 Article electronically published on November 29, 2000 GOOD IDEALS IN GORENSTEIN LOCAL RINGS SHIRO GOTO, SIN-ICHIRO IAI, AND KEI-ICHI WATANABE Abstract. Let I be an m-primary ideal in a Gorenstein local ring (A,m)with dim A = d, and assume that I contains a parameterP ideal Q in A as a reduction. n n+1 We say that I is a good ideal in A if G = n≥0 I =I is a Gorenstein ring with a(G)=1−d. The associated graded ring G of I is a Gorenstein ring with a(G)=−d if and only if I = Q. Hence good ideals in our sense are good ones next to the parameter ideals Q in A. A basic theory of good ideals is developed in this paper. We have that I is a good ideal in A if and only if I2 = QI and I = Q : I. First a criterion for finite-dimensional Gorenstein graded algebras A over fields k to have nonempty sets XA of good ideals will be given. Second in the case where d = 1 we will give a correspondence theorem between the set XA and the set YA of certain overrings of A. A characterization of good ideals in the case where d = 2 will be given in terms of the goodness in their powers. Thanks to Kato's Riemann-Roch theorem, we are able to classify the good ideals in two-dimensional Gorenstein rational local rings.
    [Show full text]