Intersection Theory in Algebraic Geometry: Counting Lines in Three Dimensional Space
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The Many Lives of the Twisted Cubic
Mathematical Assoc. of America American Mathematical Monthly 121:1 February 10, 2018 11:32 a.m. TwistedMonthly.tex page 1 The many lives of the twisted cubic Abstract. We trace some of the history of the twisted cubic curves in three-dimensional affine and projective spaces. These curves have reappeared many times in many different guises and at different times they have served as primary objects of study, as motivating examples, and as hidden underlying structures for objects considered in mathematics and its applications. 1. INTRODUCTION Because of its simple coordinate functions, the humble (affine) 3 twisted cubic curve in R , given in the parametric form ~x(t) = (t; t2; t3) (1.1) is staple of computational problems in many multivariable calculus courses. (See Fig- ure 1.) We and our students compute its intersections with planes, find its tangent vectors and lines, approximate the arclength of segments of the curve, derive its cur- vature and torsion functions, and so forth. But in many calculus books, the curve is not even identified by name and no indication of its protean nature and rich history is given. This essay is (semi-humorously) in part an attempt to remedy that situation, and (more seriously) in part a meditation on the ways that the things mathematicians study often seem to have existences of their own. Basic structures can reappear at many times in many different contexts and under different names. According to the current consensus view of mathematical historiography, historians of mathematics do well to keep these different avatars of the underlying structure separate because it is almost never correct to attribute our understanding of the sorts of connections we will be con- sidering to the thinkers of the past. -
Proofs with Perpendicular Lines
3.4 Proofs with Perpendicular Lines EEssentialssential QQuestionuestion What conjectures can you make about perpendicular lines? Writing Conjectures Work with a partner. Fold a piece of paper D in half twice. Label points on the two creases, as shown. a. Write a conjecture about AB— and CD — . Justify your conjecture. b. Write a conjecture about AO— and OB — . AOB Justify your conjecture. C Exploring a Segment Bisector Work with a partner. Fold and crease a piece A of paper, as shown. Label the ends of the crease as A and B. a. Fold the paper again so that point A coincides with point B. Crease the paper on that fold. b. Unfold the paper and examine the four angles formed by the two creases. What can you conclude about the four angles? B Writing a Conjecture CONSTRUCTING Work with a partner. VIABLE a. Draw AB — , as shown. A ARGUMENTS b. Draw an arc with center A on each To be prof cient in math, side of AB — . Using the same compass you need to make setting, draw an arc with center B conjectures and build a on each side of AB— . Label the C O D logical progression of intersections of the arcs C and D. statements to explore the c. Draw CD — . Label its intersection truth of your conjectures. — with AB as O. Write a conjecture B about the resulting diagram. Justify your conjecture. CCommunicateommunicate YourYour AnswerAnswer 4. What conjectures can you make about perpendicular lines? 5. In Exploration 3, f nd AO and OB when AB = 4 units. -
Some Intersection Theorems for Ordered Sets and Graphs
IOURNAL OF COMBINATORIAL THEORY, Series A 43, 23-37 (1986) Some Intersection Theorems for Ordered Sets and Graphs F. R. K. CHUNG* AND R. L. GRAHAM AT&T Bell Laboratories, Murray Hill, New Jersey 07974 and *Bell Communications Research, Morristown, New Jersey P. FRANKL C.N.R.S., Paris, France AND J. B. SHEARER' Universify of California, Berkeley, California Communicated by the Managing Editors Received May 22, 1984 A classical topic in combinatorics is the study of problems of the following type: What are the maximum families F of subsets of a finite set with the property that the intersection of any two sets in the family satisfies some specified condition? Typical restrictions on the intersections F n F of any F and F’ in F are: (i) FnF’# 0, where all FEF have k elements (Erdos, Ko, and Rado (1961)). (ii) IFn F’I > j (Katona (1964)). In this paper, we consider the following general question: For a given family B of subsets of [n] = { 1, 2,..., n}, what is the largest family F of subsets of [n] satsifying F,F’EF-FnFzB for some BE B. Of particular interest are those B for which the maximum families consist of so- called “kernel systems,” i.e., the family of all supersets of some fixed set in B. For example, we show that the set of all (cyclic) translates of a block of consecutive integers in [n] is such a family. It turns out rather unexpectedly that many of the results we obtain here depend strongly on properties of the well-known entropy function (from information theory). -
A Three-Dimensional Laguerre Geometry and Its Visualization
A Three-Dimensional Laguerre Geometry and Its Visualization Hans Havlicek and Klaus List, Institut fur¨ Geometrie, TU Wien 3 We describe and visualize the chains of the 3-dimensional chain geometry over the ring R("), " = 0. MSC 2000: 51C05, 53A20. Keywords: chain geometry, Laguerre geometry, affine space, twisted cubic. 1 Introduction The aim of the present paper is to discuss in some detail the Laguerre geometry (cf. [1], [6]) which arises from the 3-dimensional real algebra L := R("), where "3 = 0. This algebra generalizes the algebra of real dual numbers D = R("), where "2 = 0. The Laguerre geometry over D is the geometry on the so-called Blaschke cylinder (Figure 1); the non-degenerate conics on this cylinder are called chains (or cycles, circles). If one generator of the cylinder is removed then the remaining points of the cylinder are in one-one correspon- dence (via a stereographic projection) with the points of the plane of dual numbers, which is an isotropic plane; the chains go over R" to circles and non-isotropic lines. So the point space of the chain geometry over the real dual numbers can be considered as an affine plane with an extra “improper line”. The Laguerre geometry based on L has as point set the projective line P(L) over L. It can be seen as the real affine 3-space on L together with an “improper affine plane”. There is a point model R for this geometry, like the Blaschke cylinder, but it is more compli- cated, and belongs to a 7-dimensional projective space ([6, p. -
Projective Geometry: a Short Introduction
Projective Geometry: A Short Introduction Lecture Notes Edmond Boyer Master MOSIG Introduction to Projective Geometry Contents 1 Introduction 2 1.1 Objective . .2 1.2 Historical Background . .3 1.3 Bibliography . .4 2 Projective Spaces 5 2.1 Definitions . .5 2.2 Properties . .8 2.3 The hyperplane at infinity . 12 3 The projective line 13 3.1 Introduction . 13 3.2 Projective transformation of P1 ................... 14 3.3 The cross-ratio . 14 4 The projective plane 17 4.1 Points and lines . 17 4.2 Line at infinity . 18 4.3 Homographies . 19 4.4 Conics . 20 4.5 Affine transformations . 22 4.6 Euclidean transformations . 22 4.7 Particular transformations . 24 4.8 Transformation hierarchy . 25 Grenoble Universities 1 Master MOSIG Introduction to Projective Geometry Chapter 1 Introduction 1.1 Objective The objective of this course is to give basic notions and intuitions on projective geometry. The interest of projective geometry arises in several visual comput- ing domains, in particular computer vision modelling and computer graphics. It provides a mathematical formalism to describe the geometry of cameras and the associated transformations, hence enabling the design of computational ap- proaches that manipulates 2D projections of 3D objects. In that respect, a fundamental aspect is the fact that objects at infinity can be represented and manipulated with projective geometry and this in contrast to the Euclidean geometry. This allows perspective deformations to be represented as projective transformations. Figure 1.1: Example of perspective deformation or 2D projective transforma- tion. Another argument is that Euclidean geometry is sometimes difficult to use in algorithms, with particular cases arising from non-generic situations (e.g. -
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 ). -
HE WANG Abstract. a Mini-Course on Rational Homotopy Theory
RATIONAL HOMOTOPY THEORY HE WANG Abstract. A mini-course on rational homotopy theory. Contents 1. Introduction 2 2. Elementary homotopy theory 3 3. Spectral sequences 8 4. Postnikov towers and rational homotopy theory 16 5. Commutative differential graded algebras 21 6. Minimal models 25 7. Fundamental groups 34 References 36 2010 Mathematics Subject Classification. Primary 55P62 . 1 2 HE WANG 1. Introduction One of the goals of topology is to classify the topological spaces up to some equiva- lence relations, e.g., homeomorphic equivalence and homotopy equivalence (for algebraic topology). In algebraic topology, most of the time we will restrict to spaces which are homotopy equivalent to CW complexes. We have learned several algebraic invariants such as fundamental groups, homology groups, cohomology groups and cohomology rings. Using these algebraic invariants, we can seperate two non-homotopy equivalent spaces. Another powerful algebraic invariants are the higher homotopy groups. Whitehead the- orem shows that the functor of homotopy theory are power enough to determine when two CW complex are homotopy equivalent. A rational coefficient version of the homotopy theory has its own techniques and advan- tages: 1. fruitful algebraic structures. 2. easy to calculate. RATIONAL HOMOTOPY THEORY 3 2. Elementary homotopy theory 2.1. Higher homotopy groups. Let X be a connected CW-complex with a base point x0. Recall that the fundamental group π1(X; x0) = [(I;@I); (X; x0)] is the set of homotopy classes of maps from pair (I;@I) to (X; x0) with the product defined by composition of paths. Similarly, for each n ≥ 2, the higher homotopy group n n πn(X; x0) = [(I ;@I ); (X; x0)] n n is the set of homotopy classes of maps from pair (I ;@I ) to (X; x0) with the product defined by composition. -
Algebraic Geometry Michael Stoll
Introductory Geometry Course No. 100 351 Fall 2005 Second Part: Algebraic Geometry Michael Stoll Contents 1. What Is Algebraic Geometry? 2 2. Affine Spaces and Algebraic Sets 3 3. Projective Spaces and Algebraic Sets 6 4. Projective Closure and Affine Patches 9 5. Morphisms and Rational Maps 11 6. Curves — Local Properties 14 7. B´ezout’sTheorem 18 2 1. What Is Algebraic Geometry? Linear Algebra can be seen (in parts at least) as the study of systems of linear equations. In geometric terms, this can be interpreted as the study of linear (or affine) subspaces of Cn (say). Algebraic Geometry generalizes this in a natural way be looking at systems of polynomial equations. Their geometric realizations (their solution sets in Cn, say) are called algebraic varieties. Many questions one can study in various parts of mathematics lead in a natural way to (systems of) polynomial equations, to which the methods of Algebraic Geometry can be applied. Algebraic Geometry provides a translation between algebra (solutions of equations) and geometry (points on algebraic varieties). The methods are mostly algebraic, but the geometry provides the intuition. Compared to Differential Geometry, in Algebraic Geometry we consider a rather restricted class of “manifolds” — those given by polynomial equations (we can allow “singularities”, however). For example, y = cos x defines a perfectly nice differentiable curve in the plane, but not an algebraic curve. In return, we can get stronger results, for example a criterion for the existence of solutions (in the complex numbers), or statements on the number of solutions (for example when intersecting two curves), or classification results. -
An Introduction to Complex Algebraic Geometry with Emphasis on The
AN INTRODUCTION TO COMPLEX ALGEBRAIC GEOMETRY WITH EMPHASIS ON THE THEORY OF SURFACES By Chris Peters Mathematisch Instituut der Rijksuniversiteit Leiden and Institut Fourier Grenoble i Preface These notes are based on courses given in the fall of 1992 at the University of Leiden and in the spring of 1993 at the University of Grenoble. These courses were meant to elucidate the Mori point of view on classification theory of algebraic surfaces as briefly alluded to in [P]. The material presented here consists of a more or less self-contained advanced course in complex algebraic geometry presupposing only some familiarity with the theory of algebraic curves or Riemann surfaces. But the goal, as in the lectures, is to understand the Enriques classification of surfaces from the point of view of Mori-theory. In my opininion any serious student in algebraic geometry should be acquainted as soon as possible with the yoga of coherent sheaves and so, after recalling the basic concepts in algebraic geometry, I have treated sheaves and their cohomology theory. This part culminated in Serre’s theorems about coherent sheaves on projective space. Having mastered these tools, the student can really start with surface theory, in particular with intersection theory of divisors on surfaces. The treatment given is algebraic, but the relation with the topological intersection theory is commented on briefly. A fuller discussion can be found in Appendix 2. Intersection theory then is applied immediately to rational surfaces. A basic tool from the modern point of view is Mori’s rationality theorem. The treatment for surfaces is elementary and I borrowed it from [Wi]. -
Complete Intersection Dimension
PUBLICATIONS MATHÉMATIQUES DE L’I.H.É.S. LUCHEZAR L. AVRAMOV VESSELIN N. GASHAROV IRENA V. PEEVA Complete intersection dimension Publications mathématiques de l’I.H.É.S., tome 86 (1997), p. 67-114 <http://www.numdam.org/item?id=PMIHES_1997__86__67_0> © Publications mathématiques de l’I.H.É.S., 1997, 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 géné- rales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou im- pression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ COMPLETE INTERSECTION DIMENSION by LUGHEZAR L. AVRAMOV, VESSELIN N. GASHAROV, and IRENA V. PEEVA (1) Abstract. A new homological invariant is introduced for a finite module over a commutative noetherian ring: its CI-dimension. In the local case, sharp quantitative and structural data are obtained for modules of finite CI- dimension, providing the first class of modules of (possibly) infinite projective dimension with a rich structure theory of free resolutions. CONTENTS Introduction ................................................................................ 67 1. Homological dimensions ................................................................... 70 2. Quantum regular sequences .............................................................. -
Arxiv:1910.11630V1 [Math.AG] 25 Oct 2019 3 Geometric Invariant Theory 10 3.1 Quotients and the Notion of Stability
Geometric Invariant Theory, holomorphic vector bundles and the Harder–Narasimhan filtration Alfonso Zamora Departamento de Matem´aticaAplicada y Estad´ıstica Universidad CEU San Pablo Juli´anRomea 23, 28003 Madrid, Spain e-mail: [email protected] Ronald A. Z´u˜niga-Rojas Centro de Investigaciones Matem´aticasy Metamatem´aticas CIMM Escuela de Matem´atica,Universidad de Costa Rica UCR San Jos´e11501, Costa Rica e-mail: [email protected] Abstract. This survey intends to present the basic notions of Geometric Invariant Theory (GIT) through its paradigmatic application in the construction of the moduli space of holomorphic vector bundles. Special attention is paid to the notion of stability from different points of view and to the concept of maximal unstability, represented by the Harder-Narasimhan filtration and, from which, correspondences with the GIT picture and results derived from stratifications on the moduli space are discussed. Keywords: Geometric Invariant Theory, Harder-Narasimhan filtration, moduli spaces, vector bundles, Higgs bundles, GIT stability, symplectic stability, stratifications. MSC class: 14D07, 14D20, 14H10, 14H60, 53D30 Contents 1 Introduction 2 2 Preliminaries 4 2.1 Lie groups . .4 2.2 Lie algebras . .6 2.3 Algebraic varieties . .7 2.4 Vector bundles . .8 arXiv:1910.11630v1 [math.AG] 25 Oct 2019 3 Geometric Invariant Theory 10 3.1 Quotients and the notion of stability . 10 3.2 Hilbert-Mumford criterion . 14 3.3 Symplectic stability . 18 3.4 Examples . 21 3.5 Maximal unstability . 24 2 git, hvb & hnf 4 Moduli Space of vector bundles 28 4.1 GIT construction of the moduli space . 28 4.2 Harder-Narasimhan filtration . -
Hyperbolic Monopoles and Rational Normal Curves
HYPERBOLIC MONOPOLES AND RATIONAL NORMAL CURVES Nigel Hitchin (Oxford) Edinburgh April 20th 2009 204 Research Notes A NOTE ON THE TANGENTS OF A TWISTED CUBIC B Y M. F. ATIYAH Communicated by J. A. TODD Received 8 May 1951 1. Consider a rational normal cubic C3. In the Klein representation of the lines of $3 by points of a quadric Q in Ss, the tangents of C3 are represented by the points of a rational normal quartic O4. It is the object of this note to examine some of the consequences of this correspondence, in terms of the geometry associated with the two curves. 2. C4 lies on a Veronese surface V, which represents the congruence of chords of O3(l). Also C4 determines a 4-space 2 meeting D. in Qx, say; and since the surface of tangents of O3 is a developable, consecutive tangents intersect, and therefore the tangents to C4 lie on Q, and so on £lv Hence Qx, containing the sextic surface of tangents to C4, must be the quadric threefold / associated with C4, i.e. the quadric determining the same polarity as C4 (2). We note also that the tangents to C4 correspond in #3 to the plane pencils with vertices on O3, and lying in the corresponding osculating planes. 3. We shall prove that the surface U, which is the locus of points of intersection of pairs of osculating planes of C4, is the projection of the Veronese surface V from L, the pole of 2, on to 2. Let P denote a point of C3, and t, n the tangent line and osculating plane at P, and let T, T, w denote the same for the corresponding point of C4.