Algebraic Geometry
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1 Affine Varieties
1 Affine Varieties We will begin following Kempf's Algebraic Varieties, and eventually will do things more like in Hartshorne. We will also use various sources for commutative algebra. What is algebraic geometry? Classically, it is the study of the zero sets of polynomials. We will now fix some notation. k will be some fixed algebraically closed field, any ring is commutative with identity, ring homomorphisms preserve identity, and a k-algebra is a ring R which contains k (i.e., we have a ring homomorphism ι : k ! R). P ⊆ R an ideal is prime iff R=P is an integral domain. Algebraic Sets n n We define affine n-space, A = k = f(a1; : : : ; an): ai 2 kg. n Any f = f(x1; : : : ; xn) 2 k[x1; : : : ; xn] defines a function f : A ! k : (a1; : : : ; an) 7! f(a1; : : : ; an). Exercise If f; g 2 k[x1; : : : ; xn] define the same function then f = g as polynomials. Definition 1.1 (Algebraic Sets). Let S ⊆ k[x1; : : : ; xn] be any subset. Then V (S) = fa 2 An : f(a) = 0 for all f 2 Sg. A subset of An is called algebraic if it is of this form. e.g., a point f(a1; : : : ; an)g = V (x1 − a1; : : : ; xn − an). Exercises 1. I = (S) is the ideal generated by S. Then V (S) = V (I). 2. I ⊆ J ) V (J) ⊆ V (I). P 3. V ([αIα) = V ( Iα) = \V (Iα). 4. V (I \ J) = V (I · J) = V (I) [ V (J). Definition 1.2 (Zariski Topology). We can define a topology on An by defining the closed subsets to be the algebraic subsets. -
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. -
Using Elimination to Describe Maxwell Curves Lucas P
Louisiana State University LSU Digital Commons LSU Master's Theses Graduate School 2006 Using elimination to describe Maxwell curves Lucas P. Beverlin Louisiana State University and Agricultural and Mechanical College Follow this and additional works at: https://digitalcommons.lsu.edu/gradschool_theses Part of the Applied Mathematics Commons Recommended Citation Beverlin, Lucas P., "Using elimination to describe Maxwell curves" (2006). LSU Master's Theses. 2879. https://digitalcommons.lsu.edu/gradschool_theses/2879 This Thesis is brought to you for free and open access by the Graduate School at LSU Digital Commons. It has been accepted for inclusion in LSU Master's Theses by an authorized graduate school editor of LSU Digital Commons. For more information, please contact [email protected]. USING ELIMINATION TO DESCRIBE MAXWELL CURVES A Thesis Submitted to the Graduate Faculty of the Louisiana State University and Agricultural and Mechanical College in partial ful¯llment of the requirements for the degree of Master of Science in The Department of Mathematics by Lucas Paul Beverlin B.S., Rose-Hulman Institute of Technology, 2002 August 2006 Acknowledgments This dissertation would not be possible without several contributions. I would like to thank my thesis advisor Dr. James Madden for his many suggestions and his patience. I would also like to thank my other committee members Dr. Robert Perlis and Dr. Stephen Shipman for their help. I would like to thank my committee in the Experimental Statistics department for their understanding while I completed this project. And ¯nally I would like to thank my family for giving me the chance to achieve a higher education. -
Effective Noether Irreducibility Forms and Applications*
Appears in Journal of Computer and System Sciences, 50/2 pp. 274{295 (1995). Effective Noether Irreducibility Forms and Applications* Erich Kaltofen Department of Computer Science, Rensselaer Polytechnic Institute Troy, New York 12180-3590; Inter-Net: [email protected] Abstract. Using recent absolute irreducibility testing algorithms, we derive new irreducibility forms. These are integer polynomials in variables which are the generic coefficients of a multivariate polynomial of a given degree. A (multivariate) polynomial over a specific field is said to be absolutely irreducible if it is irreducible over the algebraic closure of its coefficient field. A specific polynomial of a certain degree is absolutely irreducible, if and only if all the corresponding irreducibility forms vanish when evaluated at the coefficients of the specific polynomial. Our forms have much smaller degrees and coefficients than the forms derived originally by Emmy Noether. We can also apply our estimates to derive more effective versions of irreducibility theorems by Ostrowski and Deuring, and of the Hilbert irreducibility theorem. We also give an effective estimate on the diameter of the neighborhood of an absolutely irreducible polynomial with respect to the coefficient space in which absolute irreducibility is preserved. Furthermore, we can apply the effective estimates to derive several factorization results in parallel computational complexity theory: we show how to compute arbitrary high precision approximations of the complex factors of a multivariate integral polynomial, and how to count the number of absolutely irreducible factors of a multivariate polynomial with coefficients in a rational function field, both in the complexity class . The factorization results also extend to the case where the coefficient field is a function field. -
Notes on Grassmannians
NOTES ON GRASSMANNIANS ANDERS SKOVSTED BUCH This is class notes under construction. We have not attempted to account for the history of the results covered here. 1. Construction of Grassmannians 1.1. The set of points. Let k = k be an algebraically closed field, and let kn be the vector space of column vectors with n coordinates. Given a non-negative integer m ≤ n, the Grassmann variety Gr(m, n) is defined as a set by Gr(m, n)= {Σ ⊂ kn | Σ is a vector subspace with dim(Σ) = m} . Our first goal is to show that Gr(m, n) has a structure of algebraic variety. 1.2. Space with functions. Let FR(n, m) = {A ∈ Mat(n × m) | rank(A) = m} be the set of all n × m matrices of full rank, and let π : FR(n, m) → Gr(m, n) be the map defined by π(A) = Span(A), the column span of A. We define a topology on Gr(m, n) be declaring the a subset U ⊂ Gr(m, n) is open if and only if π−1(U) is open in FR(n, m), and further declare that a function f : U → k is regular if and only if f ◦ π is a regular function on π−1(U). This gives Gr(m, n) the structure of a space with functions. ex:morphism Exercise 1.1. (1) The map π : FR(n, m) → Gr(m, n) is a morphism of spaces with functions. (2) Let X be a space with functions and φ : Gr(m, n) → X a map. -
Generic Flatness for Strongly Noetherian Algebras
Journal of Algebra 221, 579–610 (1999) Article ID jabr.1999.7997, available online at http://www.idealibrary.com on Generic Flatness for Strongly Noetherian Algebras M. Artin* Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 View metadata, citation and similar papersE-mail: at core.ac.uk [email protected] brought to you by CORE provided by Elsevier - Publisher Connector L. W. Small Department of Mathematics, University of California at San Diego, La Jolla, California 92093 E-mail: [email protected] and J. J. Zhang Department of Mathematics, Box 354350, University of Washington, Seattle, Washington 98195 E-mail: [email protected] Communicated by J. T. Stafford Received November 18, 1999 Key Words: closed set; critically dense; generically flat; graded ring; Nullstellen- satz; strongly noetherian; universally noetherian. © 1999 Academic Press CONTENTS 0. Introduction. 1. Blowing up a critically dense sequence. 2. A characterization of closed sets. 3. Generic flatness. 4. Strongly noetherian and universally noetherian algebras. 5. A partial converse. *All three authors were supported in part by the NSF and the third author was also supported by a Sloan Research Fellowship. 579 0021-8693/99 $30.00 Copyright © 1999 by Academic Press All rights of reproduction in any form reserved. 580 artin, small, and zhang 0. INTRODUCTION In general, A will denote a right noetherian associative algebra over a commutative noetherian ring R.LetR0 be a commutative R-algebra. If R0 is finitely generated over R, then a version of the Hilbert basis theorem 0 asserts that A ⊗R R is right noetherian. We call an algebra A strongly right 0 0 noetherian if A ⊗R R is right noetherian whenever R is noetherian. -
Etienne Bézout on Elimination Theory Erwan Penchevre
Etienne Bézout on elimination theory Erwan Penchevre To cite this version: Erwan Penchevre. Etienne Bézout on elimination theory. 2017. hal-01654205 HAL Id: hal-01654205 https://hal.archives-ouvertes.fr/hal-01654205 Preprint submitted on 3 Dec 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Etienne Bézout on elimination theory ∗ Erwan Penchèvre Bézout’s name is attached to his famous theorem. Bézout’s Theorem states that the degree of the eliminand of a system a n algebraic equations in n unknowns, when each of the equations is generic of its degree, is the product of the degrees of the equations. The eliminand is, in the terms of XIXth century algebra1, an equation of smallest degree resulting from the elimination of (n − 1) unknowns. Bézout demonstrates his theorem in 1779 in a treatise entitled Théorie générale des équations algébriques. In this text, he does not only demonstrate the theorem for n > 2 for generic equations, but he also builds a classification of equations that allows a better bound on the degree of the eliminand when the equations are not generic. This part of his work is difficult: it appears incomplete and has been seldom studied. -
Simple Proof of the Main Theorem of Elimination Theory in Algebraic Geometry
SIMPLE PROOF OF THE MAIN THEOREM OF ELIMINATION THEORY IN ALGEBRAIC GEOMETRY Autor(en): Cartier, P. / Tate, J. Objekttyp: Article Zeitschrift: L'Enseignement Mathématique Band (Jahr): 24 (1978) Heft 1-2: L'ENSEIGNEMENT MATHÉMATIQUE PDF erstellt am: 10.10.2021 Persistenter Link: http://doi.org/10.5169/seals-49707 Nutzungsbedingungen Die ETH-Bibliothek ist Anbieterin der digitalisierten Zeitschriften. Sie besitzt keine Urheberrechte an den Inhalten der Zeitschriften. Die Rechte liegen in der Regel bei den Herausgebern. Die auf der Plattform e-periodica veröffentlichten Dokumente stehen für nicht-kommerzielle Zwecke in Lehre und Forschung sowie für die private Nutzung frei zur Verfügung. Einzelne Dateien oder Ausdrucke aus diesem Angebot können zusammen mit diesen Nutzungsbedingungen und den korrekten Herkunftsbezeichnungen weitergegeben werden. Das Veröffentlichen von Bildern in Print- und Online-Publikationen ist nur mit vorheriger Genehmigung der Rechteinhaber erlaubt. Die systematische Speicherung von Teilen des elektronischen Angebots auf anderen Servern bedarf ebenfalls des schriftlichen Einverständnisses der Rechteinhaber. Haftungsausschluss Alle Angaben erfolgen ohne Gewähr für Vollständigkeit oder Richtigkeit. Es wird keine Haftung übernommen für Schäden durch die Verwendung von Informationen aus diesem Online-Angebot oder durch das Fehlen von Informationen. Dies gilt auch für Inhalte Dritter, die über dieses Angebot zugänglich sind. Ein Dienst der ETH-Bibliothek ETH Zürich, Rämistrasse 101, 8092 Zürich, Schweiz, www.library.ethz.ch http://www.e-periodica.ch A SIMPLE PROOF OF THE MAIN THEOREM OF ELIMINATION THEORY IN ALGEBRAIC GEOMETRY by P. Cartier and J. Tate Summary The purpose of this note is to provide a simple proof (which we believe to be new) for the weak zero theorem in the case of homogeneous polynomials. -
Utah/Fall 2020 3. Abstract Varieties. the Categories of Affine and Quasi
Algebraic Geometry I (Math 6130) Utah/Fall 2020 3. Abstract Varieties. The categories of affine and quasi-affine varieties over k are (by definition) full subcategories of the category of sheaved spaces (X; OX ) over k. We now enlarge the category just enough within the category of sheaved spaces to define a category of abstract varieties analogous to the categories of differentiable or analytic manifolds. Varieties are Noetherian and locally affine and separated (analogous to Hausdorff), the latter defined via products, which are also used to define the notion of a proper (analogous to compact or complete) variety. Definition 3.1. A topology on a set X is Noetherian if every descending chain: X ⊇ X1 ⊇ X2 ⊇ · · · of closed sets eventually stabilizes (or every ascending chain of open sets stabilizes). Remarks. (a) A Noetherian topological space X is quasi-compact, i.e. every open cover of X has a finite subcover, but a quasi-compact space need not be Noetherian. (b) The Zariski topology on a quasi-affine variety is Noetherian. (c) It is possible for a topological space X to fail to be Noetherian even if it has a cover by open sets, each of which is Noetherian with the induced topology. If the open cover is finite, however, then X is Noetherian. Definition 3.2. A Noetherian topological space X is reducible if X = X1 [ X2 for closed sets X1;X2 properly contained in X. Otherwise it is irreducible. Remarks. (a) As we saw in x1, every Noetherian topological space X is a finite union of irreducible closed subsets, accounting for the irreducible components of X. -
Grothendieck Ring of Varieties
Neeraja Sahasrabudhe Grothendieck Ring of Varieties Thesis advisor: J. Sebag Université Bordeaux 1 1 Contents 1. Introduction 3 2. Grothendieck Ring of Varieties 5 2.1. Classical denition 5 2.2. Classical properties 6 2.3. Bittner's denition 9 3. Stable Birational Geometry 14 4. Application 18 4.1. Grothendieck Ring is not a Domain 18 4.2. Grothendieck Ring of motives 19 5. APPENDIX : Tools for Birational geometry 20 Blowing Up 21 Resolution of Singularities 23 6. Glossary 26 References 30 1. Introduction First appeared in a letter of Grothendieck in Serre-Grothendieck cor- respondence (letter of 16/8/64), the Grothendieck ring of varieties is an interesting object lying at the heart of the theory of motivic inte- gration. A class of variety in this ring contains a lot of geometric information about the variety. For example, the topological euler characteristic, Hodge polynomials, Stably-birational properties, number of points if the variety is dened on a nite eld etc. Besides, the question of equality of these classes has given some important new results in bira- tional geometry (for example, Batyrev-Kontsevich's theorem). The Grothendieck Ring K0(V ark) is the quotient of the free abelian group generated by isomorphism classes of k-varieties, by the relation [XnY ] = [X]−[Y ], where Y is a closed subscheme of X; the ber prod- 0 0 uct over k induces a ring structure dened by [X]·[X ] = [(X ×k X )red]. Many geometric objects verify this kind of relations. It gives many re- alization maps, called additive invariants, containing some geometric information about the varieties. -
Triangle Identities Via Elimination Theory
Triangle Identities via Elimination Theory Dam Van Nhi and Luu Ba Thang Abstract The aim of this paper is to present some new identities via resultants and elimination theory, by presenting the sidelengths, the lengths of the altitudes and the exradii of a triangle as roots of cubic polynomials. 1 Introduction Finding identities between elements of a triangle represents a classical topic in elementary geometry. In Recent Advances in Geometric Inequalities [3], Mitrinovic et al presents the sidelengths, the lengths of the altitudes and the various radii as roots of cubic polynomials, and collects many interesting equalities and inequalities between these quantities. As far as we know, almost of them are built using only elementary algebra and geometry. In this paper, we use transformations of rational functions and resultants to build similar nice equalities and inequalities might be difficult to prove with different methods. 2 Main results To fix notations, suppose we are given a triangle ∆ABC with sidelengh a; b; c. Denote the radius of the circumcircle by R, the radius of incircle by r, the area of ∆ABC by S, the semiperimeter as p, the radii of the excircles as r1; r2; r3, and the altitudes from sides a; b and c, respectively, as ha; hb and hc. We have the results from [3, Chapter 1]. Theorem 1. [3] Using the above notations, (i) a; b; c are the roots of the cubic polynomial x3 − 2px2 + (p2 + r2 + 4Rr)x − 4Rrp: (1) (ii) ha; hb; hc are the roots of the cubic polynomial S2 + 4Rr3 + r4 2S2 2S2 x3 − x2 + x − : (2) 2Rr2 Rr R (iii) r1; r2; r3 are the roots of the cubic polynomial x3 − (4R + r)x2 + p2x − p2r: (3) Next, we present how to use transformations of rational functions to build new geometric equalities ax + b and inequalities. -
Completeness in Partial Differential Fields
Completeness in Partial Differential Fields James Freitag [email protected] Department of Mathematics, Statistics, and Computer Science University of Illinois at Chicago 322 Science and Engineering Offices (M/C 249) 851 S. Morgan Street Chicago, IL 60607-7045 Abstract We study completeness in partial differential varieties. We generalize many of the results of Pong to the partial differential setting. In particular, we establish a valuative criterion for differential completeness and use it to give a new class of examples of complete partial differential varieties. In addition to completeness we prove some new embedding theorems for differential algebraic varieties. We use methods from both differential algebra and model theory. Completeness is a fundamental notion in algebraic geometry. In this paper, we examine an analogue in differential algebraic geometry. The paper builds on Pong’s [16] and Kolchin’s [7] work on differential completeness in the case of differential va- rieties over ordinary differential fields and generalizes to the case differential varieties over partial differential fields with finitely many commuting derivations. Many of the proofs generalize in the straightforward manner, given that one sets up the correct definitions and attempts to prove the correct analogues of Pong’s or Kolchin’s results. arXiv:1106.0703v2 [math.LO] 3 Feb 2012 Of course, some of the results are harder to prove because our varieties may be infinite differential transcendence degree. Nevertheless, Proposition 3.3 generalizes a theorem of [17] even when we restrict to the ordinary case. The proposition also generalizes a known result projective varieties. Many of the results are model-theoretic in nature or use model-theoretic tools.