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Fundamental Algebraic Geometry
http://dx.doi.org/10.1090/surv/123 hematical Surveys and onographs olume 123 Fundamental Algebraic Geometry Grothendieck's FGA Explained Barbara Fantechi Lothar Gottsche Luc lllusie Steven L. Kleiman Nitin Nitsure AngeloVistoli American Mathematical Society U^VDED^ EDITORIAL COMMITTEE Jerry L. Bona Peter S. Landweber Michael G. Eastwood Michael P. Loss J. T. Stafford, Chair 2000 Mathematics Subject Classification. Primary 14-01, 14C20, 13D10, 14D15, 14K30, 18F10, 18D30. For additional information and updates on this book, visit www.ams.org/bookpages/surv-123 Library of Congress Cataloging-in-Publication Data Fundamental algebraic geometry : Grothendieck's FGA explained / Barbara Fantechi p. cm. — (Mathematical surveys and monographs, ISSN 0076-5376 ; v. 123) Includes bibliographical references and index. ISBN 0-8218-3541-6 (pbk. : acid-free paper) ISBN 0-8218-4245-5 (soft cover : acid-free paper) 1. Geometry, Algebraic. 2. Grothendieck groups. 3. Grothendieck categories. I Barbara, 1966- II. Mathematical surveys and monographs ; no. 123. QA564.F86 2005 516.3'5—dc22 2005053614 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. -
Intersection Theory on Regular Schemes Via Alterations and Deformation to the Normal Cone Dissertation
Intersection Theory on Regular Schemes via Alterations and Deformation to the Normal Cone Dissertation zur Erlangung des Doktorgrades der Naturwissenschaften (Dr. rer. nat.) der Fakultat¨ fur¨ Mathematik der Universitat¨ Regensburg vorgelegt von Andreas Weber aus Regensburg im Jahr 2015 Promotionsgesuch eingereicht am 13. April 2015. Die Arbeit wurde angeleitet von Prof. Dr. Klaus K¨unnemann. Pr¨ufungsausschuss: Vorsitzender: Prof. Dr. Harald Garcke 1. Gutachter: Prof. Dr. Klaus K¨unnemann 2. Gutachter: Prof. Dr. Walter Gubler weiterer Pr¨ufer: Prof. Dr. Uwe Jannsen Contents Contents 3 1 Introduction 5 2 Chow Groups of S-schemes 11 2.1 The S-Dimension . 11 2.2 Chow Groups . 14 3 Resolution of Singularities and Alterations 17 3.1 Assumption on Alterations . 17 3.2 State of the Art . 18 4 Intersection Theory with Supports on Regular Schemes 21 4.1 Bivariant Classes and Orientations . 21 4.2 Alterations and BQ-Orientations . 37 4.3 Intersection Theory with Supports on BQ-orienting Schemes . 39 5 Comparison to other Approaches to Intersection Theory 47 5.1 Intersection with Divisors . 47 5.2 Smooth Schemes over a Dedekind scheme . 49 A Fulton's Theory for S-schemes 53 A.1 Proper push-forward and flat pull-back . 53 A.2 Intersection with Divisors . 56 A.3 Cones, Chern and Segre classes . 56 A.4 Deformation to the Normal bundle . 63 A.5 Refined Gysin homomorphisms . 65 A.6 Intersection theory for smooth schemes over a one-dimensional base . 69 Bibliography 71 Chapter 1 Introduction k For a Noetherian separated regular scheme X, the Chow group CHY (X) of algebraic cycles of codimension k with supports in a closed subset Y of X is given as k k k CHY (X) := ZY (X) = RatY (X); k i.e. -
Waring-Type Problems for Polynomials Algebra Meets Geometry Alessandro Oneto
Waring-type problems for polynomials Algebra meets Geometry Alessandro Oneto Waring-type problems for polynomials Algebra meets Geometry Alessandro Oneto ©Alessandro Oneto, Stockholm University 2016 e-mail: [email protected] ISBN: 978-91-7649-424-0 Printed by Holmbergs, Malmö 2016 Distributor: Department of Mathematics, Stockholm University CONTENTS 1 INTRODUCTION 3 1.1 ADDITIVE DECOMPOSITIONS OF INTEGERS ............ 3 1.2 ADDITIVE DECOMPOSITIONS OF POLYNOMIALS .......... 4 1.2.1 CLASSICAL WARING DECOMPOSITIONS .......... 5 1.2.2 d-TH WARING DECOMPOSITIONS .............. 7 1.2.3 WARING-LIKE DECOMPOSITIONS .............. 8 1.2.4 REAL WARING DECOMPOSITIONS ............. 9 1.3 GEOMETRIC INTERPRETATION ................... 10 1.3.1 SECANT VARIETIES ..................... 10 1.3.2 CLASSICAL WARING PROBLEM:VERONESE VARIETIES . 12 1.3.3 d-TH WARING PROBLEM: VARIETIES OF POWERS . 14 1.3.4 WARING-LIKE PROBLEMS: VARIETIES OF µ-POWERS . 17 1.3.5 TERRACINI’S LEMMA .................... 18 2 APOLARITY THEORY AND POINTS CONFIGURATIONS 23 2.1 APOLARITY THEORY ......................... 23 2.2 HILBERT FUNCTIONS OF CONFIGURATIONS OF REDUCED POINTS 27 2.3 WARING LOCI OF HOMOGENEOUS POLYNOMIALS . 29 2.3.1 QUADRICS .......................... 31 2.3.2 MONOMIALS ......................... 32 2.3.3 BINARY FORMS ....................... 34 2.3.4 PLANE CUBICS ....................... 37 2.4 WARING LOCI AND THE STRASSEN CONJECTURE . 43 2.5 APOLARITY LEMMA: POWER IDEALS AND FAT POINTS . 49 2.5.1 IDEALS OF FAT POINTS ................... 49 2.5.2 INVERSE SYSTEMS OF IDEALS FAT POINTS. 51 2.6 HILBERT FUNCTIONS OF CONFIGURATIONS OF FAT POINTS . 53 2.6.1 DOUBLE POINTS: THE ALEXANDER–HIRSCHOWITZ THEO- REM ............................. 55 2.7 SPECIAL CONFIGURATIONS OF FAT POINTS . -
An Estimate of Canonical Dimension of Groups Based on Schubert Calculus
An estimate of canonical dimension of groups based on Schubert calculus Rostislav Devyatov∗ November 3, 2020 Abstract We sketch the proof of a connection between the canonical (0-)dimension of semisimple split simply connected groups and cohomology of their full flag varieties. Using this connection, we get a new estimate of the canonical (0-)dimension of simply connected split exceptional groups of type E understood as a group. A full proof will be published later. 1 Introduction To define the canonical (0-)dimension of an algebraic group understood as a group, we first need to define the canonical (0-)dimension of a scheme understood as a scheme (which is a different definition). Roughly speaking, the canonical (0-)dimension of a scheme is a number indicating how hard it is to get a rational point in the scheme. The canonical (0-)dimension of an algebraic group shows how hard it is to get rational points in torsors related to the group. To be more precise, let us fix some conventions and give some definitions. We speak of algebraic schemes and use stacks project as the source of basic definitions. All schemes in the present text are of finite type over a field and separated. The base field is arbitrary. Speaking of canonical dimension of schemes, there are two closely related notions in the literature: the canonical 0-dimension of a scheme defined in [14] and the canonical dimension of a scheme defined in [9]. These two definitions are not known to be always equivalent, but they are equivalent for two particular classes of schemes: for smooth complete schemes and for torsors of split reductive groups (see [13, Theorem 1.16, Remark 1.17, and Example 1.18]). -
Intersection Theory
APPENDIX A Intersection Theory In this appendix we will outline the generalization of intersection theory and the Riemann-Roch theorem to nonsingular projective varieties of any dimension. To motivate the discussion, let us look at the case of curves and surfaces, and then see what needs to be generalized. For a divisor D on a curve X, leaving out the contribution of Serre duality, we can write the Riemann-Roch theorem (IV, 1.3) as x(.!Z'(D)) = deg D + 1 - g, where xis the Euler characteristic (III, Ex. 5.1). On a surface, we can write the Riemann-Roch theorem (V, 1.6) as 1 x(!l'(D)) = 2 D.(D - K) + 1 + Pa· In each case, on the left-hand side we have something involving cohomol ogy groups of the sheaf !l'(D), while on the right-hand side we have some numerical data involving the divisor D, the canonical divisor K, and some invariants of the variety X. Of course the ultimate aim of a Riemann-Roch type theorem is to compute the dimension of the linear system IDI or of lnDI for large n (II, Ex. 7.6). This is achieved by combining a formula for x(!l'(D)) with some vanishing theorems for Hi(X,!l'(D)) fori > 0, such as the theorems of Serre (III, 5.2) or Kodaira (III, 7.15). We will now generalize these results so as to give an expression for x(!l'(D)) on a nonsingular projective variety X of any dimension. And while we are at it, with no extra effort we get a formula for x(t&"), where @" is any coherent locally free sheaf. -
18.726 Algebraic Geometry Spring 2009
MIT OpenCourseWare http://ocw.mit.edu 18.726 Algebraic Geometry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.726: Algebraic Geometry (K.S. Kedlaya, MIT, Spring 2009) More properties of schemes (updated 9 Mar 09) I’ve now spent a fair bit of time discussing properties of morphisms of schemes. How ever, there are a few properties of individual schemes themselves that merit some discussion (especially for those of you interested in arithmetic applications); here are some of them. 1 Reduced schemes I already mentioned the notion of a reduced scheme. An affine scheme X = Spec(A) is reduced if A is a reduced ring (i.e., A has no nonzero nilpotent elements). This occurs if and only if each stalk Ap is reduced. We say X is reduced if it is covered by reduced affine schemes. Lemma. Let X be a scheme. The following are equivalent. (a) X is reduced. (b) For every open affine subsheme U = Spec(R) of X, R is reduced. (c) For each x 2 X, OX;x is reduced. Proof. A previous exercise. Recall that any closed subset Z of a scheme X supports a unique reduced closed sub- scheme, defined by the ideal sheaf I which on an open affine U = Spec(A) is defined by the intersection of the prime ideals p 2 Z \ U. See Hartshorne, Example 3.2.6. 2 Connected schemes A nonempty scheme is connected if its underlying topological space is connected, i.e., cannot be written as a disjoint union of two open sets. -
K-Stability of Relative Flag Varieties
K-stability of relative flag varieties Anton Isopoussu University of Cambridge Department of Pure Mathematics and Mathematical Statistics Churchill College arXiv:1307.7638v2 [math.AG] 10 Nov 2015 April 2015 This dissertation is submitted for the degree of Doctor of Philosophy Abstract We generalise partial results about the Yau-Tian-Donaldson correspondence on ruled manifolds to bundles whose fibre is a classical flag variety. This is done using Chern class computations involving the combinatorics of Schur functors. The strongest results are obtained when working over a Riemann surface. Weaker partial results are obtained for adiabatic polarisations in the general case. We develop the notion of relative K-stability which embeds the idea of working over a base variety into the theory of K-stability. We equip the set of equivalence classes of test configuration with the structure of a convex space fibred over the cone of rational polarisations. From this, we deduce the open- ness of the K-unstable locus. We illustrate our new algebraic constructions with several examples. Declaration This dissertation is the result of my own work and includes nothing which is the outcome of work done in collaboration except as declared in the Preface and specified in the text. It is not substantially the same as any that I have submitted, or, is being concurrently submitted for a degree or diploma or other qualification at the University of Cambridge or any other University or similar institution except as declared in the Preface and specified in the text. I further state that no substantial part of my dissertation has already been submitted, or, is being concurrently submitted for any such degree, diploma or other qualification at the University of Cambridge or any other University of similar institution except as specified in the text Anton Isopoussu November 11, 2015 Acknowledgements I gratefully acknowledge the patient guidance of my supervisor Dr. -
Foundations of Algebraic Geometry Class 37
FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 37 RAVI VAKIL CONTENTS 1. Motivation and game plan 1 2. The affine case: three definitions 2 Welcome back to the third quarter! The theme for this quarter, insofar as there is one, will be “useful ideas to know”. We'll start with differentials for the first three lectures. I prefer to start any topic with a number of examples, but in this case I'm going to spend a fair amount of time discussing technicalities, and then get to a number of exam- ples. Here is the main message I want you to get. Differentials are an intuitive geometric notion, and we're going to figure out the right description of them algebraically. I find the algebraic manifestation a little non-intuitive, so I always like to tie it to the geometry. So please don't tune out of the statements. Also, I want you to notice that although the algebraic statements are odd, none of the proofs are hard or long. This topic could have been done as soon as we knew about morphisms and quasico- herent sheaves. 1. MOTIVATION AND GAME PLAN Suppose X is a “smooth” k-variety. We hope to define a tangent bundle. We'll see that the right way to do this will easily apply in much more general circumstances. • We'll see that cotangent is more “natural” for schemes than tangent bundle. This is similar to the fact that the Zariski cotangent space is more natural than the tangent space (i.e. if A is a ring and m is a maximal ideal, then m=m2 is “more natural” than (m=m2)_. -
Foundations of Algebraic Geometry Class 1
FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 1 CONTENTS 1. Welcome 1 2. Why algebraic geometry? 3 3. Preliminaries on category theory 5 3.1. Functors 5 3.2. Universal properties 6 3.3. Yoneda's Lemma 7 Today: About this course. Why algebraic geometry? Motivation and program. Crash course in category theory: universal properties, Yoneda's lemma. 1. WELCOME Welcome! This is Math 216A, Foundations of Algebraic Geometry, the first of a three- quarter sequence on the topic. I'd like to tell you a little about what I intend with this course. Algebraic geometry is a subject that somehow connects and unifies several parts of mathematics, including obviously algebra and geometry, but also number theory, and depending on your point of view many other things, including topology, string theory, etc. As a result, it can be a handy thing to know if you are in a variety of subjects, no- tably number theory, symplectic geometry, and certain kinds of topology. The power of the field arises from a point of view that was developed in the 1960's in Paris, by the group led by Alexandre Grothendieck. The power comes from rather heavy formal and technical machinery, in which it is easy to lose sight of the intuitive nature of the objects under consideration. This is one reason why it used to strike fear into the hearts of the uninitiated. The rough edges have been softened over the ensuing decades, but there is an in- escapable need to understand the subject on its own terms. This class is intended to be an experiment. -
Foundations of Algebraic Geometry Classes 43 and 44
FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASSES 43 AND 44 RAVI VAKIL CONTENTS 1. Flat implies constant Euler characteristic 1 2. Proof of Important Theorem on constancy of Euler characteristic in flat families 5 3. Start of Thursday's class: Review 5 4. Cohomology and base change theorems 6 5. Line bundles are trivial in a Zariski-closed locus, and glimpses of the relative Picard scheme 9 This week: constancy of Euler characteristic in flat families. The semicontinuity theorem and consequences. Glimpses of the relative Picard scheme. 1. FLAT IMPLIES CONSTANT EULER CHARACTERISTIC We come to an important consequence of flatness. We'll see that this result implies many answers and examples to questions that we would have asked before we even knew about flatness. 1.1. Important Theorem. — Suppose f : X Y is a projective morphism, and F is a coherent ! i i sheaf on X, flat over Y. Suppose Y is locally Noetherian. Then P(-1) h (XyFjy) is a locally constant function of y 2 Y. In other words, the Euler characteristic of F is constant in the fibers. This is first sign that cohomology behaves well in families. (We'll soon see a second: the Semicontinuity Theorem 4.4.) Before getting to the proof, I'll show you some of its many consequences. (A second proof will be given after the semicontinuity discussion.) The theorem also gives a necessary condition for flatness. It also sufficient if target is integral and locally Noetherian, although we won't use this. (Reference: You can trans- late Hartshorne Theorem III.9.9 into this.) I seem to recall that both the necessary and sufficient conditions are due to Serre, but I'm not sure of a reference. -
Descent of Morphisms
Descent of Morphisms Pol van Hoften University of Utrecht, 4001613 [email protected] March 29, 2016 1 Introduction and Statement of the Main Theorem Today we will prove that schemes are sheaves in the ´etaletopology. However, with a little additional work we can show something more general, namely that schemes are sheaves in the fpqc topology. Clearly, we will first have to define the fpqc topology and show that all ´etalecovers are fpqc-covers. The fpqc topology is sometimes defined incorrectly, so to avoid confusion I have included the incorrect definition: Definition 1. (WRONG!) A morphism of schemes f : X ! Y is called quasi-compact if Y can be −1 covered by affine open subsets fVigi2I such that f (Vi) is quasi-compact, for all i 2 I. A morphisms S of schemes fUi ! V gi2I is called an fpqc cover is the map i2I ! Y is flat, surjective and quasi- compact. This seems like a perfectly fine definition, except that Zariski open covers are not fpqc covers! Indeed, the inclusion of an open subscheme U ! X doesn't need to be quasi-compact (even if X is affine! See [Vak] Exercise 3.6.G (b)). The correct definition is more complicated: Proposition 1 (Proposition 2.33 [Vis05]). Let f : X ! Y be a surjective morphism of schemes, then the following are equivalent 1. Every quasi-compact open subset of Y is the image of a quasi-compact open subset of X. 2. There exists a covering fVig of Y by open affine subschemes, such that each Vi is the image of a quasi-compact open subset of X. -
SUMMER 2016 ALGEBRAIC GEOMETRY SEMINAR Contents 1. Separability, Varieties and Rational Maps: 5/16/16 1 2. Proper Morphisms
SUMMER 2016 ALGEBRAIC GEOMETRY SEMINAR ARUN DEBRAY JULY 28, 2016 Contents 1. Separability, Varieties and Rational Maps: 5/16/161 2. Proper Morphisms: 5/19/166 3. Dimension: 5/23/16 10 4. Codimension One: 5/26/16 12 5. Regularity: 5/30/16 14 6. An Algebraic Interlude: 6/1/16 15 7. More Regularity and Smoothness: 6/2/16 17 8. Quasicoherent Sheaves: 6/6/16 20 9. Coherent Sheaves: 6/9/16 23 10. Line Bundles: 6/13/16 24 11. Effective Cartier Divisors and Closed Subschemes: 6/16/16 27 12. Quasicoherent Sheaves on Projective Schemes: 6/20/16 30 13. Finite Type Quasicoherent Sheaves on Projective Schemes: 6/23/16 33 14. Pushforwards and Pullbacks of Quasicoherent Sheaves: 6/27/16 34 15. Relative Spec and Proj: 7/11/16 37 16. Nice Results About Curves: 7/14/16 41 17. Cechˇ Cohomology: 7/18/16 43 18. Cechˇ Cohomology, II: 7/21/16 46 19. Curves: 7/25/16 48 20. Elliptic and Hyperelliptic Curves: 7/28/16 51 1. Separability, Varieties and Rational Maps: 5/16/16 Today's lecture was given by Tom Oldfield, on the first half of chapter 10. This seminar has a website, located at https://www.ma.utexas.edu/users/toldfield/Seminars/Algebraicgeometryreading.html. The first half of Chapter 10 is about separated morphisms and varieties; it only took us 10 chapters! Vakil writes that he was very conflicted about leaving a proper treatment of algebraic varieties, a cornerstone of classical algebraic geometry, to so late in the notes.