Hilton, Plane Algebraic Curves
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Polar Coordinates and Complex Numbers Infinite Series Vectors and Matrices Motion Along a Curve Partial Derivatives
Contents CHAPTER 9 Polar Coordinates and Complex Numbers 9.1 Polar Coordinates 348 9.2 Polar Equations and Graphs 351 9.3 Slope, Length, and Area for Polar Curves 356 9.4 Complex Numbers 360 CHAPTER 10 Infinite Series 10.1 The Geometric Series 10.2 Convergence Tests: Positive Series 10.3 Convergence Tests: All Series 10.4 The Taylor Series for ex, sin x, and cos x 10.5 Power Series CHAPTER 11 Vectors and Matrices 11.1 Vectors and Dot Products 11.2 Planes and Projections 11.3 Cross Products and Determinants 11.4 Matrices and Linear Equations 11.5 Linear Algebra in Three Dimensions CHAPTER 12 Motion along a Curve 12.1 The Position Vector 446 12.2 Plane Motion: Projectiles and Cycloids 453 12.3 Tangent Vector and Normal Vector 459 12.4 Polar Coordinates and Planetary Motion 464 CHAPTER 13 Partial Derivatives 13.1 Surfaces and Level Curves 472 13.2 Partial Derivatives 475 13.3 Tangent Planes and Linear Approximations 480 13.4 Directional Derivatives and Gradients 490 13.5 The Chain Rule 497 13.6 Maxima, Minima, and Saddle Points 504 13.7 Constraints and Lagrange Multipliers 514 CHAPTER 12 Motion Along a Curve I [ 12.1 The Position Vector I-, This chapter is about "vector functions." The vector 2i +4j + 8k is constant. The vector R(t) = ti + t2j+ t3k is moving. It is a function of the parameter t, which often represents time. At each time t, the position vector R(t) locates the moving body: position vector =R(t) =x(t)i + y(t)j + z(t)k. -
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. -
INTRODUCTION to ALGEBRAIC GEOMETRY, CLASS 25 Contents 1
INTRODUCTION TO ALGEBRAIC GEOMETRY, CLASS 25 RAVI VAKIL Contents 1. The genus of a nonsingular projective curve 1 2. The Riemann-Roch Theorem with Applications but No Proof 2 2.1. A criterion for closed immersions 3 3. Recap of course 6 PS10 back today; PS11 due today. PS12 due Monday December 13. 1. The genus of a nonsingular projective curve The definition I’m going to give you isn’t the one people would typically start with. I prefer to introduce this one here, because it is more easily computable. Definition. The tentative genus of a nonsingular projective curve C is given by 1 − deg ΩC =2g 2. Fact (from Riemann-Roch, later). g is always a nonnegative integer, i.e. deg K = −2, 0, 2,.... Complex picture: Riemann-surface with g “holes”. Examples. Hence P1 has genus 0, smooth plane cubics have genus 1, etc. Exercise: Hyperelliptic curves. Suppose f(x0,x1) is a polynomial of homo- geneous degree n where n is even. Let C0 be the affine plane curve given by 2 y = f(1,x1), with the generically 2-to-1 cover C0 → U0.LetC1be the affine 2 plane curve given by z = f(x0, 1), with the generically 2-to-1 cover C1 → U1. Check that C0 and C1 are nonsingular. Show that you can glue together C0 and C1 (and the double covers) so as to give a double cover C → P1. (For computational convenience, you may assume that neither [0; 1] nor [1; 0] are zeros of f.) What goes wrong if n is odd? Show that the tentative genus of C is n/2 − 1.(Thisisa special case of the Riemann-Hurwitz formula.) This provides examples of curves of any genus. -
1915-16.] the " Geometria Organica" of Colin Maclaurin. 87 V.—-The
1915-16.] The " Geometria Organica" of Colin Maclaurin. 87 V.—-The " Geometria Organica " of Colin Maclaurin: A Historical and Critical Survey. By Charles Tweedie, M.A., B.Sc, Lecturer in Mathematics, Edinburgh University. (MS. received October 15, 1915. Bead December 6,1915.) INTRODUCTION. COLIN MACLATJBIN, the celebrated mathematician, was born in 1698 at Kilmodan in Argyllshire, where his father was minister of the parish. In 1709 he entered Glasgow University, where his mathematical talent rapidly developed under the fostering care of Professor Robert Simson. In 171*7 he successfully competed for the Chair of Mathematics in the Marischal College of Aberdeen University. In 1719 he came directly under the personal influence of Newton, when on a visit to London, bearing with him the manuscript of the Geometria Organica, published in quarto in 1720. The publication of this work immediately brought him into promi- nence in the scientific world. In 1725 he was, on the recommendation of Newton, elected to the Chair of Mathematics in Edinburgh University, which he occupied until his death in 1746. As a lecturer Maclaurin was a conspicuous success. He took great pains to make his subject as clear and attractive as possible, so much so that he made mathematics " a fashionable study." The labour of teaching his numerous students seriously curtailed the time he could spare for original research. In quantity his works do not bulk largely, but what he did produce was, in the main, of superlative quality, presented clearly and concisely. The Geometria Organica and the Geometrical Appendix to his Treatise on Algebra give him a place in the first rank of great geometers, forming as they do the basis of the theory of the Higher Plane Curves; while his Treatise of Fluxions (1742) furnished an unassailable bulwark and text-book for the study of the Calculus. -
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 ). -
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. -
Orthocorrespondence and Orthopivotal Cubics
Forum Geometricorum b Volume 3 (2003) 1–27. bbb FORUM GEOM ISSN 1534-1178 Orthocorrespondence and Orthopivotal Cubics Bernard Gibert Abstract. We define and study a transformation in the triangle plane called the orthocorrespondence. This transformation leads to the consideration of a fam- ily of circular circumcubics containing the Neuberg cubic and several hitherto unknown ones. 1. The orthocorrespondence Let P be a point in the plane of triangle ABC with barycentric coordinates (u : v : w). The perpendicular lines at P to AP , BP, CP intersect BC, CA, AB respectively at Pa, Pb, Pc, which we call the orthotraces of P . These orthotraces 1 lie on a line LP , which we call the orthotransversal of P . We denote the trilinear ⊥ pole of LP by P , and call it the orthocorrespondent of P . A P P ∗ P ⊥ B C Pa Pc LP H/P Pb Figure 1. The orthotransversal and orthocorrespondent In barycentric coordinates, 2 ⊥ 2 P =(u(−uSA + vSB + wSC )+a vw : ··· : ···), (1) Publication Date: January 21, 2003. Communicating Editor: Paul Yiu. We sincerely thank Edward Brisse, Jean-Pierre Ehrmann, and Paul Yiu for their friendly and valuable helps. 1The homography on the pencil of lines through P which swaps a line and its perpendicular at P is an involution. According to a Desargues theorem, the points are collinear. 2All coordinates in this paper are homogeneous barycentric coordinates. Often for triangle cen- ters, we list only the first coordinate. The remaining two can be easily obtained by cyclically permut- ing a, b, c, and corresponding quantities. Thus, for example, in (1), the second and third coordinates 2 2 are v(−vSB + wSC + uSA)+b wu and w(−wSC + uSA + vSB )+c uv respectively. -
Arxiv:2009.05223V1 [Math.NT] 11 Sep 2020 Fdegree of Eaeitrse Nfidn Function a finding in Interested Are We Hoe 1.1
COUNTING ELLIPTIC CURVES WITH A RATIONAL N-ISOGENY FOR SMALL N BRANDON BOGGESS AND SOUMYA SANKAR Abstract. We count the number of rational elliptic curves of bounded naive height that have a rational N-isogeny, for N ∈ {2, 3, 4, 5, 6, 8, 9, 12, 16, 18}. For some N, this is done by generalizing a method of Harron and Snowden. For the remaining cases, we use the framework of Ellenberg, Satriano and Zureick-Brown, in which the naive height of an elliptic curve is the height of the corresponding point on a moduli stack. 1. Introduction ′ Let E be an elliptic curve over Q. An isogeny φ : E E between two elliptic curves is said to be cyclic ¯ → of degree N if Ker(φ)(Q) ∼= Z/NZ. Further, it is said to be rational if Ker(φ) is stable under the action of the absolute Galois group, GQ. A natural question one can ask is, how many elliptic curves over Q have a rational cyclic N-isogeny? Henceforth, we will omit the adjective ‘cyclic’, since these are the only types of isogenies we will consider. It is classically known that for N 10 and N = 12, 13, 16, 18, 25, there are infinitely many such elliptic curves. Thus we order them by naive≤ height. An elliptic curve E over Q has a unique minimal Weierstrass equation y2 = x3 + Ax + B where A, B Z and gcd(A3,B2) is not divisible by any 12th power. Define the naive height of E to be ht(E) = max A∈3, B 2 . -
Curvature Theorems on Polar Curves
PROCEEDINGS OF THE NATIONAL ACADEMY OF SCIENCES Volume 33 March 15, 1947 Number 3 CURVATURE THEOREMS ON POLAR CUR VES* By EDWARD KASNER AND JOHN DE CICCO DEPARTMENT OF MATHEMATICS, COLUMBIA UNIVERSITY, NEW YoRK Communicated February 4, 1947 1. The principle of duality in projective geometry is based on the theory of poles and polars with respect to a conic. For a conic, a point has only one kind of polar, the first polar, or polar straight line. However, for a general algebraic curve of higher degree n, a point has not only the first polar of degree n - 1, but also the second polar of degree n - 2, the rth polar of degree n - r, :. ., the (n - 1) polar of degree 1. This last polar is a straight line. The general polar theory' goes back to Newton, and was developed by Bobillier, Cayley, Salmon, Clebsch, Aronhold, Clifford and Mayer. For a given curve Cn of degree n, the first polar of any point 0 of the plane is a curve C.-1 of degree n - 1. If 0 is a point of C, it is well known that the polar curve C,,- passes through 0 and touches the given curve Cn at 0. However, the two curves do not have the same curvature. We find that the ratio pi of the curvature of C,- i to that of Cn is Pi = (n -. 2)/(n - 1). For example, if the given curve is a cubic curve C3, the first polar is a conic C2, and at an ordinary point 0 of C3, the ratio of the curvatures is 1/2. -
Some Curves and the Lengths of Their Arcs Amelia Carolina Sparavigna
Some Curves and the Lengths of their Arcs Amelia Carolina Sparavigna To cite this version: Amelia Carolina Sparavigna. Some Curves and the Lengths of their Arcs. 2021. hal-03236909 HAL Id: hal-03236909 https://hal.archives-ouvertes.fr/hal-03236909 Preprint submitted on 26 May 2021 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. Some Curves and the Lengths of their Arcs Amelia Carolina Sparavigna Department of Applied Science and Technology Politecnico di Torino Here we consider some problems from the Finkel's solution book, concerning the length of curves. The curves are Cissoid of Diocles, Conchoid of Nicomedes, Lemniscate of Bernoulli, Versiera of Agnesi, Limaçon, Quadratrix, Spiral of Archimedes, Reciprocal or Hyperbolic spiral, the Lituus, Logarithmic spiral, Curve of Pursuit, a curve on the cone and the Loxodrome. The Versiera will be discussed in detail and the link of its name to the Versine function. Torino, 2 May 2021, DOI: 10.5281/zenodo.4732881 Here we consider some of the problems propose in the Finkel's solution book, having the full title: A mathematical solution book containing systematic solutions of many of the most difficult problems, Taken from the Leading Authors on Arithmetic and Algebra, Many Problems and Solutions from Geometry, Trigonometry and Calculus, Many Problems and Solutions from the Leading Mathematical Journals of the United States, and Many Original Problems and Solutions. -
Generalization of the Pedal Concept in Bidimensional Spaces. Application to the Limaçon of Pascal• Generalización Del Concep
Generalization of the pedal concept in bidimensional spaces. Application to the limaçon of Pascal• Irene Sánchez-Ramos a, Fernando Meseguer-Garrido a, José Juan Aliaga-Maraver a & Javier Francisco Raposo-Grau b a Escuela Técnica Superior de Ingeniería Aeronáutica y del Espacio, Universidad Politécnica de Madrid, Madrid, España. [email protected], [email protected], [email protected] b Escuela Técnica Superior de Arquitectura, Universidad Politécnica de Madrid, Madrid, España. [email protected] Received: June 22nd, 2020. Received in revised form: February 4th, 2021. Accepted: February 19th, 2021. Abstract The concept of a pedal curve is used in geometry as a generation method for a multitude of curves. The definition of a pedal curve is linked to the concept of minimal distance. However, an interesting distinction can be made for . In this space, the pedal curve of another curve C is defined as the locus of the foot of the perpendicular from the pedal point P to the tangent2 to the curve. This allows the generalization of the definition of the pedal curve for any given angle that is not 90º. ℝ In this paper, we use the generalization of the pedal curve to describe a different method to generate a limaçon of Pascal, which can be seen as a singular case of the locus generation method and is not well described in the literature. Some additional properties that can be deduced from these definitions are also described. Keywords: geometry; pedal curve; distance; angularity; limaçon of Pascal. Generalización del concepto de curva podal en espacios bidimensionales. Aplicación a la Limaçon de Pascal Resumen El concepto de curva podal está extendido en la geometría como un método generativo para multitud de curvas. -
Rational Curve with One Cusp
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 89, Number I, September 1983 RATIONAL CURVE WITH ONE CUSP HI SAO YOSHIHARA Abstract. In this paper we shall give new examples of plane curves C satisfying C — {P} = A1 for some point fECby using automorphisms of open surfaces. 1. In this paper we assume the ground field is an algebraically closed field of characteristic zero. Let C be an irreducible algebraic curve in P2. Then let us call C a curve of type I if C — {P} = A1 for some point PEC, and a curve of type II if C\L = Ax for some line L. Of course a curve of type II is of type I. Curves of type II have been studied by Abhyankar and Moh [1] and others. On the other hand, if the logarithmic Kodaira dimension of P2 — C is -oo or the automorphism group of P2 —■C is infinite dimensional, then C is of type I [3,6]. So the study of type I seems to be important, but only a few examples are known that are of type I and not of type II. Moreover the definitions of those examples are not so plain [4,5]. Here we shall give new examples by using automorphisms of open surfaces. 2. First we recall the properties of type I. Let (ex,...,et) be the sequence of the multiplicities of the infinitely near singular points of P in case C is of type I with degree d > 3. Then put t R = d2- 2 e2-e, + 1. Since a curve of type II is an image of a line by some automorphism of A2 [1], we see that R > 2 for this curve by the same argument as below.