The Geometry of Schemes
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Questions About Boij-S\" Oderberg Theory
QUESTIONS ABOUT BOIJ–SODERBERG¨ THEORY DANIEL ERMAN AND STEVEN V SAM 1. Background on Boij–Soderberg¨ Theory Boij–S¨oderberg theory focuses on the properties and duality relationship between two types of numerical invariants. One side involves the Betti table of a graded free resolution over the polynomial ring. The other side involves the cohomology table of a coherent sheaf on projective space. The theory began with a conjectural description of the cone of Betti tables of finite length modules, given in [10]. Those conjectures were proven in [25], which also described the cone of cohomology tables of vector bundles and illustrated a sort of duality between Betti tables and cohomology tables. The theory itself has since expanded in many directions: allowing modules whose support has higher dimension, replacing vector bundles by coherent sheaves, working over rings other than the polynomial ring, and so on. But at its core, Boij–S¨oderberg theory involves: (1) A classification, up to scalar multiple, of the possible Betti tables of some class of objects (for example, free resolutions of finitely generated modules of dimension ≤ c). (2) A classification, up to scalar multiple, of the cohomology tables of some class of objects (for examples, coherent sheaves of dimension ≤ n − c). (3) Intersection theory-style duality results between Betti tables and cohomology tables. One motivation behind Boij and S¨oderberg’s conjectures was the observation that it would yield an immediate proof of the Cohen–Macaulay version of the Multiplicity Conjectures of Herzog–Huneke–Srinivasan [44]. Eisenbud and Schreyer’s [25] thus yielded an immediate proof of that conjecture, and the subsequent papers [11, 26] provided a proof of the Mul- tiplicity Conjecture for non-Cohen–Macaulay modules. -
Modules and Vector Spaces
Modules and Vector Spaces R. C. Daileda October 16, 2017 1 Modules Definition 1. A (left) R-module is a triple (R; M; ·) consisting of a ring R, an (additive) abelian group M and a binary operation · : R × M ! M (simply written as r · m = rm) that for all r; s 2 R and m; n 2 M satisfies • r(m + n) = rm + rn ; • (r + s)m = rm + sm ; • r(sm) = (rs)m. If R has unity we also require that 1m = m for all m 2 M. If R = F , a field, we call M a vector space (over F ). N Remark 1. One can show that as a consequence of this definition, the zeros of R and M both \act like zero" relative to the binary operation between R and M, i.e. 0Rm = 0M and r0M = 0M for all r 2 R and m 2 M. H Example 1. Let R be a ring. • R is an R-module using multiplication in R as the binary operation. • Every (additive) abelian group G is a Z-module via n · g = ng for n 2 Z and g 2 G. In fact, this is the only way to make G into a Z-module. Since we must have 1 · g = g for all g 2 G, one can show that n · g = ng for all n 2 Z. Thus there is only one possible Z-module structure on any abelian group. • Rn = R ⊕ R ⊕ · · · ⊕ R is an R-module via | {z } n times r(a1; a2; : : : ; an) = (ra1; ra2; : : : ; ran): • Mn(R) is an R-module via r(aij) = (raij): • R[x] is an R-module via X i X i r aix = raix : i i • Every ideal in R is an R-module. -
Field Theory Pete L. Clark
Field Theory Pete L. Clark Thanks to Asvin Gothandaraman and David Krumm for pointing out errors in these notes. Contents About These Notes 7 Some Conventions 9 Chapter 1. Introduction to Fields 11 Chapter 2. Some Examples of Fields 13 1. Examples From Undergraduate Mathematics 13 2. Fields of Fractions 14 3. Fields of Functions 17 4. Completion 18 Chapter 3. Field Extensions 23 1. Introduction 23 2. Some Impossible Constructions 26 3. Subfields of Algebraic Numbers 27 4. Distinguished Classes 29 Chapter 4. Normal Extensions 31 1. Algebraically closed fields 31 2. Existence of algebraic closures 32 3. The Magic Mapping Theorem 35 4. Conjugates 36 5. Splitting Fields 37 6. Normal Extensions 37 7. The Extension Theorem 40 8. Isaacs' Theorem 40 Chapter 5. Separable Algebraic Extensions 41 1. Separable Polynomials 41 2. Separable Algebraic Field Extensions 44 3. Purely Inseparable Extensions 46 4. Structural Results on Algebraic Extensions 47 Chapter 6. Norms, Traces and Discriminants 51 1. Dedekind's Lemma on Linear Independence of Characters 51 2. The Characteristic Polynomial, the Trace and the Norm 51 3. The Trace Form and the Discriminant 54 Chapter 7. The Primitive Element Theorem 57 1. The Alon-Tarsi Lemma 57 2. The Primitive Element Theorem and its Corollary 57 3 4 CONTENTS Chapter 8. Galois Extensions 61 1. Introduction 61 2. Finite Galois Extensions 63 3. An Abstract Galois Correspondence 65 4. The Finite Galois Correspondence 68 5. The Normal Basis Theorem 70 6. Hilbert's Theorem 90 72 7. Infinite Algebraic Galois Theory 74 8. A Characterization of Normal Extensions 75 Chapter 9. -
Bibliography
Bibliography [1] Emil Artin. Galois Theory. Dover, second edition, 1964. [2] Michael Artin. Algebra. Prentice Hall, first edition, 1991. [3] M. F. Atiyah and I. G. Macdonald. Introduction to Commutative Algebra. Addison Wesley, third edition, 1969. [4] Nicolas Bourbaki. Alg`ebre, Chapitres 1-3.El´ements de Math´ematiques. Hermann, 1970. [5] Nicolas Bourbaki. Alg`ebre, Chapitre 10.El´ements de Math´ematiques. Masson, 1980. [6] Nicolas Bourbaki. Alg`ebre, Chapitres 4-7.El´ements de Math´ematiques. Masson, 1981. [7] Nicolas Bourbaki. Alg`ebre Commutative, Chapitres 8-9.El´ements de Math´ematiques. Masson, 1983. [8] Nicolas Bourbaki. Elements of Mathematics. Commutative Algebra, Chapters 1-7. Springer–Verlag, 1989. [9] Henri Cartan and Samuel Eilenberg. Homological Algebra. Princeton Math. Series, No. 19. Princeton University Press, 1956. [10] Jean Dieudonn´e. Panorama des mat´ematiques pures. Le choix bourbachique. Gauthiers-Villars, second edition, 1979. [11] David S. Dummit and Richard M. Foote. Abstract Algebra. Wiley, second edition, 1999. [12] Albert Einstein. Zur Elektrodynamik bewegter K¨orper. Annalen der Physik, 17:891–921, 1905. [13] David Eisenbud. Commutative Algebra With A View Toward Algebraic Geometry. GTM No. 150. Springer–Verlag, first edition, 1995. [14] Jean-Pierre Escofier. Galois Theory. GTM No. 204. Springer Verlag, first edition, 2001. [15] Peter Freyd. Abelian Categories. An Introduction to the theory of functors. Harper and Row, first edition, 1964. [16] Sergei I. Gelfand and Yuri I. Manin. Homological Algebra. Springer, first edition, 1999. [17] Sergei I. Gelfand and Yuri I. Manin. Methods of Homological Algebra. Springer, second edition, 2003. [18] Roger Godement. Topologie Alg´ebrique et Th´eorie des Faisceaux. -
Arxiv:1912.10980V2 [Math.AG] 28 Jan 2021 6
Automorphisms of real del Pezzo surfaces and the real plane Cremona group Egor Yasinsky* Universität Basel Departement Mathematik und Informatik Spiegelgasse 1, 4051 Basel, Switzerland ABSTRACT. We study automorphism groups of real del Pezzo surfaces, concentrating on finite groups acting with invariant Picard number equal to one. As a result, we obtain a vast part of classification of finite subgroups in the real plane Cremona group. CONTENTS 1. Introduction 2 1.1. The classification problem2 1.2. G-surfaces3 1.3. Some comments on the conic bundle case4 1.4. Notation and conventions6 2. Some auxiliary results7 2.1. A quick look at (real) del Pezzo surfaces7 2.2. Sarkisov links8 2.3. Topological bounds9 2.4. Classical linear groups 10 3. Del Pezzo surfaces of degree 8 10 4. Del Pezzo surfaces of degree 6 13 5. Del Pezzo surfaces of degree 5 16 arXiv:1912.10980v2 [math.AG] 28 Jan 2021 6. Del Pezzo surfaces of degree 4 18 6.1. Topology and equations 18 6.2. Automorphisms 20 6.3. Groups acting minimally on real del Pezzo quartics 21 7. Del Pezzo surfaces of degree 3: cubic surfaces 28 Sylvester non-degenerate cubic surfaces 34 7.1. Clebsch diagonal cubic 35 *[email protected] Keywords: Cremona group, conic bundle, del Pezzo surface, automorphism group, real algebraic surface. 1 2 7.2. Cubic surfaces with automorphism group S4 36 Sylvester degenerate cubic surfaces 37 7.3. Equianharmonic case: Fermat cubic 37 7.4. Non-equianharmonic case 39 7.5. Non-cyclic Sylvester degenerate surfaces 39 8. Del Pezzo surfaces of degree 2 40 9. -
The Geometry of Syzygies
The Geometry of Syzygies A second course in Commutative Algebra and Algebraic Geometry David Eisenbud University of California, Berkeley with the collaboration of Freddy Bonnin, Clement´ Caubel and Hel´ ene` Maugendre For a current version of this manuscript-in-progress, see www.msri.org/people/staff/de/ready.pdf Copyright David Eisenbud, 2002 ii Contents 0 Preface: Algebra and Geometry xi 0A What are syzygies? . xii 0B The Geometric Content of Syzygies . xiii 0C What does it mean to solve linear equations? . xiv 0D Experiment and Computation . xvi 0E What’s In This Book? . xvii 0F Prerequisites . xix 0G How did this book come about? . xix 0H Other Books . 1 0I Thanks . 1 0J Notation . 1 1 Free resolutions and Hilbert functions 3 1A Hilbert’s contributions . 3 1A.1 The generation of invariants . 3 1A.2 The study of syzygies . 5 1A.3 The Hilbert function becomes polynomial . 7 iii iv CONTENTS 1B Minimal free resolutions . 8 1B.1 Describing resolutions: Betti diagrams . 11 1B.2 Properties of the graded Betti numbers . 12 1B.3 The information in the Hilbert function . 13 1C Exercises . 14 2 First Examples of Free Resolutions 19 2A Monomial ideals and simplicial complexes . 19 2A.1 Syzygies of monomial ideals . 23 2A.2 Examples . 25 2A.3 Bounds on Betti numbers and proof of Hilbert’s Syzygy Theorem . 26 2B Geometry from syzygies: seven points in P3 .......... 29 2B.1 The Hilbert polynomial and function. 29 2B.2 . and other information in the resolution . 31 2C Exercises . 34 3 Points in P2 39 3A The ideal of a finite set of points . -
Arxiv:1708.06494V1 [Math.AG] 22 Aug 2017 Proof
CLOSED POINTS ON SCHEMES JUSTIN CHEN Abstract. This brief note gives a survey on results relating to existence of closed points on schemes, including an elementary topological characterization of the schemes with (at least one) closed point. X Let X be a topological space. For a subset S ⊆ X, let S = S denote the closure of S in X. Recall that a topological space is sober if every irreducible closed subset has a unique generic point. The following is well-known: Proposition 1. Let X be a Noetherian sober topological space, and x ∈ X. Then {x} contains a closed point of X. Proof. If {x} = {x} then x is a closed point. Otherwise there exists x1 ∈ {x}\{x}, so {x} ⊇ {x1}. If x1 is not a closed point, then continuing in this way gives a descending chain of closed subsets {x} ⊇ {x1} ⊇ {x2} ⊇ ... which stabilizes to a closed subset Y since X is Noetherian. Then Y is the closure of any of its points, i.e. every point of Y is generic, so Y is irreducible. Since X is sober, Y is a singleton consisting of a closed point. Since schemes are sober, this shows in particular that any scheme whose under- lying topological space is Noetherian (e.g. any Noetherian scheme) has a closed point. In general, it is of basic importance to know that a scheme has closed points (or not). For instance, recall that every affine scheme has a closed point (indeed, this is equivalent to the axiom of choice). In this direction, one can give a simple topological characterization of the schemes with closed points. -
Real Rank Two Geometry Arxiv:1609.09245V3 [Math.AG] 5
Real Rank Two Geometry Anna Seigal and Bernd Sturmfels Abstract The real rank two locus of an algebraic variety is the closure of the union of all secant lines spanned by real points. We seek a semi-algebraic description of this set. Its algebraic boundary consists of the tangential variety and the edge variety. Our study of Segre and Veronese varieties yields a characterization of tensors of real rank two. 1 Introduction Low-rank approximation of tensors is a fundamental problem in applied mathematics [3, 6]. We here approach this problem from the perspective of real algebraic geometry. Our goal is to give an exact semi-algebraic description of the set of tensors of real rank two and to characterize its boundary. This complements the results on tensors of non-negative rank two presented in [1], and it offers a generalization to the setting of arbitrary varieties, following [2]. A familiar example is that of 2 × 2 × 2-tensors (xijk) with real entries. Such a tensor lies in the closure of the real rank two tensors if and only if the hyperdeterminant is non-negative: 2 2 2 2 2 2 2 2 x000x111 + x001x110 + x010x101 + x011x100 + 4x000x011x101x110 + 4x001x010x100x111 −2x000x001x110x111 − 2x000x010x101x111 − 2x000x011x100x111 (1) −2x001x010x101x110 − 2x001x011x100x110 − 2x010x011x100x101 ≥ 0: If this inequality does not hold then the tensor has rank two over C but rank three over R. To understand this example geometrically, consider the Segre variety X = Seg(P1 × P1 × P1), i.e. the set of rank one tensors, regarded as points in the projective space P7 = 2 2 2 7 arXiv:1609.09245v3 [math.AG] 5 Apr 2017 P(C ⊗ C ⊗ C ). -
Geometry on Grassmannians and Applications to Splitting Bundles and Smoothing Cycles
PUBLICATIONS MATHÉMATIQUES DE L’I.H.É.S. STEVEN L. KLEIMAN Geometry on grassmannians and applications to splitting bundles and smoothing cycles Publications mathématiques de l’I.H.É.S., tome 36 (1969), p. 281-297. <http://www.numdam.org/item?id=PMIHES_1969__36__281_0> © Publications mathématiques de l’I.H.É.S., 1969, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » (http://www. ihes.fr/IHES/Publications/Publications.html), implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fi- chier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ GEOMETRY ON GRASSMANNIANS AND APPLICATIONS TO SPLITTING BUNDLES AND SMOOTHING CYCLES by STEVEN L. KLEIMAN (1) INTRODUCTION Let k be an algebraically closed field, V a smooth ^-dimensional subscheme of projective space over A:, and consider the following problems: Problem 1 (splitting bundles). — Given a (vector) bundle G on V, find a monoidal transformation f \ V'->V with smooth center CT such thatyG contains a line bundle P. Problem 2 (smoothing cycles). — Given a cycle Z on V, deform Z by rational equivalence into the difference Z^ — Zg of two effective cycles whose prime components are all smooth. Strengthened form. — Given any finite number of irreducible subschemes V, of V, choose a (resp. Z^, Zg) such that for all i, the intersection V^n a (resp. -
3 Lecture 3: Spectral Spaces and Constructible Sets
3 Lecture 3: Spectral spaces and constructible sets 3.1 Introduction We want to analyze quasi-compactness properties of the valuation spectrum of a commutative ring, and to do so a digression on constructible sets is needed, especially to define the notion of constructibility in the absence of noetherian hypotheses. (This is crucial, since perfectoid spaces will not satisfy any kind of noetherian condition in general.) The reason that this generality is introduced in [EGA] is for the purpose of proving openness and closedness results on the locus of fibers satisfying reasonable properties, without imposing noetherian assumptions on the base. One first proves constructibility results on the base, often by deducing it from constructibility on the source and applying Chevalley’s theorem on images of constructible sets (which is valid for finitely presented morphisms), and then uses specialization criteria for constructible sets to be open. For our purposes, the role of constructibility will be quite different, resting on the interesting “constructible topology” that is introduced in [EGA, IV1, 1.9.11, 1.9.12] but not actually used later in [EGA]. This lecture is organized as follows. We first deal with the constructible topology on topological spaces. We discuss useful characterizations of constructibility in the case of spectral spaces, aiming for a criterion of Hochster (see Theorem 3.3.9) which will be our tool to show that Spv(A) is spectral, our ultimate goal. Notational convention. From now on, we shall write everywhere (except in some definitions) “qc” for “quasi-compact”, “qs” for “quasi-separated”, and “qcqs” for “quasi-compact and quasi-separated”. -
ALGEBRAIC GEOMETRY (PART III) EXAMPLE SHEET 3 (1) Let X Be a Scheme and Z a Closed Subset of X. Show That There Is a Closed Imme
ALGEBRAIC GEOMETRY (PART III) EXAMPLE SHEET 3 CAUCHER BIRKAR (1) Let X be a scheme and Z a closed subset of X. Show that there is a closed immersion f : Y ! X which is a homeomorphism of Y onto Z. (2) Show that a scheme X is affine iff there are b1; : : : ; bn 2 OX (X) such that each D(bi) is affine and the ideal generated by all the bi is OX (X) (see [Hartshorne, II, exercise 2.17]). (3) Let X be a topological space and let OX to be the constant sheaf associated to Z. Show that (X; OX ) is a ringed space and that every sheaf on X is in a natural way an OX -module. (4) Let (X; OX ) be a ringed space. Show that the kernel and image of a morphism of OX -modules are also OX -modules. Show that the direct sum, direct product, direct limit and inverse limit of OX -modules are OX -modules. If F and G are O H F G O L X -modules, show that omOX ( ; ) is also an X -module. Finally, if is a O F F O subsheaf of an X -module , show that the quotient sheaf L is also an X - module. O F O O F ' (5) Let (X; X ) be a ringed space and an X -module. Prove that HomOX ( X ; ) F(X). (6) Let f :(X; OX ) ! (Y; OY ) be a morphism of ringed spaces. Show that for any O F O G ∗G F ' G F X -module and Y -module we have HomOX (f ; ) HomOY ( ; f∗ ). -
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.