Alexander Duality in Intersection Theory Compositio Mathematica, Tome 70, No 3 (1989), P
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Bézout's Theorem
Bézout's theorem Toni Annala Contents 1 Introduction 2 2 Bézout's theorem 4 2.1 Ane plane curves . .4 2.2 Projective plane curves . .6 2.3 An elementary proof for the upper bound . 10 2.4 Intersection numbers . 12 2.5 Proof of Bézout's theorem . 16 3 Intersections in Pn 20 3.1 The Hilbert series and the Hilbert polynomial . 20 3.2 A brief overview of the modern theory . 24 3.3 Intersections of hypersurfaces in Pn ...................... 26 3.4 Intersections of subvarieties in Pn ....................... 32 3.5 Serre's Tor-formula . 36 4 Appendix 42 4.1 Homogeneous Noether normalization . 42 4.2 Primes associated to a graded module . 43 1 Chapter 1 Introduction The topic of this thesis is Bézout's theorem. The classical version considers the number of points in the intersection of two algebraic plane curves. It states that if we have two algebraic plane curves, dened over an algebraically closed eld and cut out by multi- variate polynomials of degrees n and m, then the number of points where these curves intersect is exactly nm if we count multiple intersections and intersections at innity. In Chapter 2 we dene the necessary terminology to state this theorem rigorously, give an elementary proof for the upper bound using resultants, and later prove the full Bézout's theorem. A very modest background should suce for understanding this chapter, for example the course Algebra II given at the University of Helsinki should cover most, if not all, of the prerequisites. In Chapter 3, we generalize Bézout's theorem to higher dimensional projective spaces. -
The Historical Development of Algebraic Geometry∗
The Historical Development of Algebraic Geometry∗ Jean Dieudonn´e March 3, 1972y 1 The lecture This is an enriched transcription of footage posted by the University of Wis- consin{Milwaukee Department of Mathematical Sciences [1]. The images are composites of screenshots from the footage manipulated with Python and the Python OpenCV library [2]. These materials were prepared by Ryan C. Schwiebert with the goal of capturing and restoring some of the material. The lecture presents the content of Dieudonn´e'sarticle of the same title [3] as well as the content of earlier lecture notes [4], but the live lecture is natu- rally embellished with different details. The text presented here is, for the most part, written as it was spoken. This is why there are some ungrammatical con- structions of spoken word, and some linguistic quirks of a French speaker. The transcriber has taken some liberties by abbreviating places where Dieudonn´e self-corrected. That is, short phrases which turned out to be false starts have been elided, and now only the subsequent word-choices that replaced them ap- pear. Unfortunately, time has taken its toll on the footage, and it appears that a brief portion at the beginning, as well as the last enumerated period (VII. Sheaves and schemes), have been lost. (Fortunately, we can still read about them in [3]!) The audio and video quality has contributed to several transcription errors. Finally, a lack of thorough knowledge of the history of algebraic geometry has also contributed some errors. Contact with suggestions for improvement of arXiv:1803.00163v1 [math.HO] 1 Mar 2018 the materials is welcome. -
INTERSECTION NUMBERS, TRANSFERS, and GROUP ACTIONS by Daniel H. Gottlieb and Murad Ozaydin Purdue University University of Oklah
INTERSECTION NUMBERS, TRANSFERS, AND GROUP ACTIONS by Daniel H. Gottlieb and Murad Ozaydin Purdue University University of Oklahoma Abstract We introduce a new definition of intersection number by means of umkehr maps. This definition agrees with an obvious definition up to sign. It allows us to extend the concept parametrically to fibre bundles. For fibre bundles we can then define intersection number transfers. Fibre bundles arise naturally in equivariant situations, so the trace of the action divides intersection numbers. We apply this to equivariant projective varieties and other examples. Also we study actions of non-connected groups and their traces. AMS subject classification 1991: Primary: 55R12, 57S99, 55M25 Secondary: 14L30 Keywords: Poincare Duality, trace of action, Sylow subgroups, Euler class, projective variety. 1 1. Introduction When two manifolds with complementary dimensions inside a third manifold intersect transversally, their intersection number is defined. This classical topological invariant has played an important role in topology and its applications. In this paper we introduce a new definition of intersection number. The definition naturally extends to the case of parametrized manifolds over a base B; that is to fibre bundles over B . Then we can define transfers for the fibre bundles related to the intersection number. This is our main objective , and it is done in Theorem 6. The method of producing transfers depends upon the Key Lemma which we prove here. Special cases of this lemma played similar roles in the establishment of the Euler–Poincare transfer, the Lefschetz number transfer, and the Nakaoka transfer. Two other features of our definition are: 1) We replace the idea of submanifolds with the idea of maps from manifolds, and in effect we are defining the intersection numbers of the maps; 2) We are not restricted to only two maps, we can define the intersection number for several maps, or submanifolds. -
Positivity in Algebraic Geometry I
Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics 48 Positivity in Algebraic Geometry I Classical Setting: Line Bundles and Linear Series Bearbeitet von R.K. Lazarsfeld 1. Auflage 2004. Buch. xviii, 387 S. Hardcover ISBN 978 3 540 22533 1 Format (B x L): 15,5 x 23,5 cm Gewicht: 1650 g Weitere Fachgebiete > Mathematik > Geometrie > Elementare Geometrie: Allgemeines Zu Inhaltsverzeichnis schnell und portofrei erhältlich bei Die Online-Fachbuchhandlung beck-shop.de ist spezialisiert auf Fachbücher, insbesondere Recht, Steuern und Wirtschaft. Im Sortiment finden Sie alle Medien (Bücher, Zeitschriften, CDs, eBooks, etc.) aller Verlage. Ergänzt wird das Programm durch Services wie Neuerscheinungsdienst oder Zusammenstellungen von Büchern zu Sonderpreisen. Der Shop führt mehr als 8 Millionen Produkte. Introduction to Part One Linear series have long stood at the center of algebraic geometry. Systems of divisors were employed classically to study and define invariants of pro- jective varieties, and it was recognized that varieties share many properties with their hyperplane sections. The classical picture was greatly clarified by the revolutionary new ideas that entered the field starting in the 1950s. To begin with, Serre’s great paper [530], along with the work of Kodaira (e.g. [353]), brought into focus the importance of amplitude for line bundles. By the mid 1960s a very beautiful theory was in place, showing that one could recognize positivity geometrically, cohomologically, or numerically. During the same years, Zariski and others began to investigate the more complicated be- havior of linear series defined by line bundles that may not be ample. -
Arxiv:1710.06183V2 [Math.AG]
ALGEBRAIC INTEGRABILITY OF FOLIATIONS WITH NUMERICALLY TRIVIAL CANONICAL BUNDLE ANDREAS HÖRING AND THOMAS PETERNELL Abstract. Given a reflexive sheaf on a mildly singular projective variety, we prove a flatness criterion under certain stability conditions. This implies the algebraicity of leaves for sufficiently stable foliations with numerically trivial canonical bundle such that the second Chern class does not vanish. Combined with the recent work of Druel and Greb-Guenancia-Kebekus this establishes the Beauville-Bogomolov decomposition for minimal models with trivial canon- ical class. 1. introduction 1.A. Main result. Let X be a normal complex projective variety that is smooth in codimension two, and let E be a reflexive sheaf on X. If E is slope-stable with respect to some ample divisor H of slope µH (E)=0, then a famous result of Mehta-Ramanathan [MR84] says that the restriction EC to a general complete intersection C of sufficiently ample divisors is stable and nef. On the other hand the variety X contains many dominating families of irreducible curves to which Mehta-Ramanathan does not apply; therefore one expects that, apart from very special situations, EC will not be nef for many curves C. If E is locally free, denote by π : P(E) → X the projectivisation of E and by ζ := c1(OP(E)(1)) the tautological class on P(E). The nefness of EC then translates into the nefness of the restriction of ζ to P(EC). Thus, the stability of E implies some positivity of the tautological class ζ. On the other hand, the non-nefness of EC on many curves can be rephrased by saying that the tautological class ζ should not be pseudoeffective. -
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. -
A Note on the Homotopy Type of the Alexander Dual
A NOTE ON THE HOMOTOPY TYPE OF THE ALEXANDER DUAL EL´IAS GABRIEL MINIAN AND JORGE TOMAS´ RODR´IGUEZ Abstract. We investigate the homotopy type of the Alexander dual of a simplicial complex. In general the homotopy type of K does not determine the homotopy type of its dual K∗. Moreover, one can construct for each finitely presented group G, a simply ∗ connected simplicial complex K such that π1(K ) = G. We study sufficient conditions on K for K∗ to have the homotopy type of a sphere. We also extend the simplicial Alexander duality to the context of reduced lattices. 1. Introduction Let A be a compact and locally contractible proper subspace of Sn. The classical n Alexander duality theorem asserts that the reduced homology groups Hi(S − A) are isomorphic to the reduced cohomology groups Hn−i−1(A) (see for example [6, Thm. 3.44]). The combinatorial (or simplicial) Alexander duality is a special case of the classical duality: if K is a finite simplicial complex and K∗ is the Alexander dual with respect to a ground set V ⊇ K0, then for any i ∼ n−i−3 ∗ Hi(K) = H (K ): Here K0 denotes the set of vertices (i.e. 0-simplices) of K and n is the size of V .A very nice and simple proof of the combinatorial Alexander duality can be found in [4]. An alternative proof of this combinatorial duality can be found in [3]. In these notes we relate the homotopy type of K with that of K∗. We show first that, even though the homology of K determines the homology of K∗ (and vice versa), the homotopy type of K does not determine the homotopy type of K∗. -
Notation and Basic Facts in Knot Theory
Appendix A Notation and Basic Facts in Knot Theory In this appendix, our aim is to provide a quick review of basic terminology and some facts in knot theory, which we use or need in this book. This appendix lists them item by item (without proof); so we do no attempt to give a full rigorous treatments; Instead, we work somewhat intuitively. The reader with interest in the details could consult the textbooks [BZ, R, Lic, KawBook]. • First, we fix notation on the circle S1 := {(x, y) ∈ R2 | x2 + y2 = 1}, and consider a finite disjoint union S1. A link is a C∞-embedding of L :S1 → S3 in the 3-sphere. We denote often by #L the number of the disjoint union, and denote the image Im(L) by only L for short. If #L is 1, L is usually called a knot, and is written K instead. This book discusses embeddings together with orientation. For an oriented link L, we denote by −L the link with its orientation reversed, and by L∗ the mirror image of L. • (Notations of link components). Given a link L :S1 → S3, let us fix an open tubular neighborhood νL ⊂ S3. Throughout this book, we denote the complement S3 \ νL by S3 \ L for short. Since we mainly discuss isotopy classes of S3 \ L,we may ignore the choice of νL. 2 • For example, for integers s, t ∈ Z , the torus link Ts,t (of type (s, t)) is defined by 3 2 s t 2 2 S Ts,t := (z,w)∈ C z + w = 0, |z| +|w| = 1 . -
Arxiv:1607.08163V2 [Math.GT] 24 Sep 2016
LECTURES ON THE TRIANGULATION CONJECTURE CIPRIAN MANOLESCU Abstract. We outline the proof that non-triangulable manifolds exist in any dimension greater than four. The arguments involve homology cobordism invariants coming from the Pinp2q symmetry of the Seiberg-Witten equations. We also explore a related construction, of an involutive version of Heegaard Floer homology. 1. Introduction The triangulation conjecture stated that every topological manifold can be triangulated. The work of Casson [AM90] in the 1980's provided counterexamples in dimension 4. The main purpose of these notes is to describe the proof of the following theorem. Theorem 1.1 ([Man13]). There exist non-triangulable n-dimensional topological manifolds for every n ¥ 5. The proof relies on previous work by Galewski-Stern [GS80] and Matumoto [Mat78], who reduced this problem to a different one, about the homology cobordism group in three dimensions. Homology cobordism can be explored using the techniques of gauge theory, as was done, for example, by Fintushel and Stern [FS85, FS90], Furuta [Fur90], and Frøyshov [Fro10]. In [Man13], Pinp2q-equivariant Seiberg-Witten Floer homology is used to construct three new invariants of homology cobordism, called α, β and γ. The properties of β suffice to answer the question raised by Galewski-Stern and Matumoto, and thus prove Theorem 1.1. The paper is organized as follows. Section 2 contains background material about triangulating manifolds. In particular, we sketch the arguments of Galewski-Stern and Matumoto that reduced Theorem 1.1 to a problem about homology cobordism. In Section 3 we introduce the Seiberg-Witten equations, finite dimensional approxima- tion, and the Conley index. -
The Ways I Know How to Define the Alexander Polynomial
ALL THE WAYS I KNOW HOW TO DEFINE THE ALEXANDER POLYNOMIAL KYLE A. MILLER Abstract. These began as notes for a talk given at the Student 3-manifold seminar, Spring 2019. There seems to be many ways to define the Alexander polynomial, all of which are somehow interrelated, but sometimes there is not an obvious path between any two given definitions. As the title suggests, this is an exploration of all the ways I know how to define this knot invariant. While we will touch on a number of facts and properties, these notes are not meant to be a complete survey of the Alexander polynomial. Contents 1. Introduction2 2. Alexander’s definition2 2.1. The Dehn presentation2 2.2. Abelianization4 2.3. The associated matrix4 3. The Alexander modules6 3.1. Orders7 3.2. Elementary ideals8 3.3. The Fox calculus 10 3.4. Torus knots 13 3.5. The Wirtinger presentation 13 3.6. Generalization to links 15 3.7. Fibered knots 15 3.8. Seifert presentation 16 3.9. Duality 18 4. The Conway potential 18 4.1. Alexander’s three-term relation 20 5. The HOMFLY-PT polynomial 24 6. Kauffman state sum 26 6.1. Another state sum 29 7. The Burau representation 29 8. Vassiliev invariants 29 9. The Alexander quandle 29 9.1. Projective Alexander quandles 33 10. Reidemeister torsion 33 11. Knot Floer homology 33 References 33 Date: May 3, 2019. 1 DRAFT 2020/11/14 18:57:28 2 KYLE A. MILLER 1. Introduction Recall that a link is an embedded closed 1-manifold in S3, and a knot is a 1-component link. -
Positive Alexander Duality for Pursuit and Evasion
POSITIVE ALEXANDER DUALITY FOR PURSUIT AND EVASION ROBERT GHRIST AND SANJEEVI KRISHNAN Abstract. Considered is a class of pursuit-evasion games, in which an evader tries to avoid detection. Such games can be formulated as the search for sections to the com- plement of a coverage region in a Euclidean space over a timeline. Prior results give homological criteria for evasion in the general case that are not necessary and sufficient. This paper provides a necessary and sufficient positive cohomological criterion for evasion in a general case. The principal tools are (1) a refinement of the Cechˇ cohomology of a coverage region with a positive cone encoding spatial orientation, (2) a refinement of the Borel-Moore homology of the coverage gaps with a positive cone encoding time orienta- tion, and (3) a positive variant of Alexander Duality. Positive cohomology decomposes as the global sections of a sheaf of local positive cohomology over the time axis; we show how this decomposition makes positive cohomology computable as a linear program. 1. Introduction The motivation for this paper comes from a type of pursuit-evasion game. In such games, two classes of agents, pursuers and evaders move in a fixed geometric domain over time. The goal of a pursuer is to capture an evader (by, e.g., physical proximity or line-of-sight). The goal of an evader is to move in such a manner so as to avoid capture by any pursuer. This paper solves a feasibility problem of whether an evader can win in a particular setting under certain constraints. We specialize to the setting of pursuers-as-sensors, in which, at each time, a certain region of space is \sensed" and any evader in this region is considered captured. -
Intersections of Hypersurfaces and Ring of Conditions of a Spherical Homogeneous Space
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 16 (2020), 016, 12 pages Intersections of Hypersurfaces and Ring of Conditions of a Spherical Homogeneous Space Kiumars KAVEH y and Askold G. KHOVANSKII zx y Department of Mathematics, University of Pittsburgh, Pittsburgh, PA, USA E-mail: [email protected] z Department of Mathematics, University of Toronto, Toronto, Canada E-mail: [email protected] x Moscow Independent University, Moscow, Russia Received November 04, 2019, in final form March 14, 2020; Published online March 20, 2020 https://doi.org/10.3842/SIGMA.2020.016 Abstract. We prove a version of the BKK theorem for the ring of conditions of a spherical homogeneous space G=H. We also introduce the notion of ring of complete intersections, firstly for a spherical homogeneous space and secondly for an arbitrary variety. Similarly to the ring of conditions of the torus, the ring of complete intersections of G=H admits a description in terms of volumes of polytopes. Key words: BKK theorem; spherical variety; Newton{Okounkov polytope; ring of conditions 2020 Mathematics Subject Classification: 14M27; 14M25; 14M10 1 Introduction ∗ n Let Γ1;:::; Γn be a set of n hypersurfaces in the n-dimensional complex torus (C ) defined by equations P1 = 0;:::;Pn = 0 where P1;:::;Pn are generic Laurent polynomials with given Newton polyhedra ∆1;:::; ∆n. The Bernstein{Koushnirenko{Khovanskii theorem (or BKK the- orem, see [2, 16, 18]) claims that the intersection number of the hypersurfaces Γ1;:::; Γn is equal to the mixed volume of ∆1;:::; ∆n multiplied by n!. In particular when ∆1 = ··· = ∆n = ∆ this theorem can be regarded as giving a formula for the degree of an ample divisor class in a projective toric variety.