Algebraic Geometry Michael Stoll
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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. -
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. -
Trigonometry Cram Sheet
Trigonometry Cram Sheet August 3, 2016 Contents 6.2 Identities . 8 1 Definition 2 7 Relationships Between Sides and Angles 9 1.1 Extensions to Angles > 90◦ . 2 7.1 Law of Sines . 9 1.2 The Unit Circle . 2 7.2 Law of Cosines . 9 1.3 Degrees and Radians . 2 7.3 Law of Tangents . 9 1.4 Signs and Variations . 2 7.4 Law of Cotangents . 9 7.5 Mollweide’s Formula . 9 2 Properties and General Forms 3 7.6 Stewart’s Theorem . 9 2.1 Properties . 3 7.7 Angles in Terms of Sides . 9 2.1.1 sin x ................... 3 2.1.2 cos x ................... 3 8 Solving Triangles 10 2.1.3 tan x ................... 3 8.1 AAS/ASA Triangle . 10 2.1.4 csc x ................... 3 8.2 SAS Triangle . 10 2.1.5 sec x ................... 3 8.3 SSS Triangle . 10 2.1.6 cot x ................... 3 8.4 SSA Triangle . 11 2.2 General Forms of Trigonometric Functions . 3 8.5 Right Triangle . 11 3 Identities 4 9 Polar Coordinates 11 3.1 Basic Identities . 4 9.1 Properties . 11 3.2 Sum and Difference . 4 9.2 Coordinate Transformation . 11 3.3 Double Angle . 4 3.4 Half Angle . 4 10 Special Polar Graphs 11 3.5 Multiple Angle . 4 10.1 Limaçon of Pascal . 12 3.6 Power Reduction . 5 10.2 Rose . 13 3.7 Product to Sum . 5 10.3 Spiral of Archimedes . 13 3.8 Sum to Product . 5 10.4 Lemniscate of Bernoulli . 13 3.9 Linear Combinations . -
Study of Spiral Transition Curves As Related to the Visual Quality of Highway Alignment
A STUDY OF SPIRAL TRANSITION CURVES AS RELA'^^ED TO THE VISUAL QUALITY OF HIGHWAY ALIGNMENT JERRY SHELDON MURPHY B, S., Kansas State University, 1968 A MJvSTER'S THESIS submitted in partial fulfillment of the requirements for the degree MASTER OF SCIENCE Department of Civil Engineering KANSAS STATE UNIVERSITY Manhattan, Kansas 1969 Approved by P^ajQT Professor TV- / / ^ / TABLE OF CONTENTS <2, 2^ INTRODUCTION 1 LITERATURE SEARCH 3 PURPOSE 5 SCOPE 6 • METHOD OF SOLUTION 7 RESULTS 18 RECOMMENDATIONS FOR FURTHER RESEARCH 27 CONCLUSION 33 REFERENCES 34 APPENDIX 36 LIST OF TABLES TABLE 1, Geonetry of Locations Studied 17 TABLE 2, Rates of Change of Slope Versus Curve Ratings 31 LIST OF FIGURES FIGURE 1. Definition of Sight Distance and Display Angle 8 FIGURE 2. Perspective Coordinate Transformation 9 FIGURE 3. Spiral Curve Calculation Equations 12 FIGURE 4. Flow Chart 14 FIGURE 5, Photograph and Perspective of Selected Location 15 FIGURE 6. Effect of Spiral Curves at Small Display Angles 19 A, No Spiral (Circular Curve) B, Completely Spiralized FIGURE 7. Effects of Spiral Curves (DA = .015 Radians, SD = 1000 Feet, D = l** and A = 10*) 20 Plate 1 A. No Spiral (Circular Curve) B, Spiral Length = 250 Feet FIGURE 8. Effects of Spiral Curves (DA = ,015 Radians, SD = 1000 Feet, D = 1° and A = 10°) 21 Plate 2 A. Spiral Length = 500 Feet B. Spiral Length = 1000 Feet (Conpletely Spiralized) FIGURE 9. Effects of Display Angle (D = 2°, A = 10°, Ig = 500 feet, = SD 500 feet) 23 Plate 1 A. Display Angle = .007 Radian B. Display Angle = .027 Radiaji FIGURE 10. -
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 . -
The Ordered Distribution of Natural Numbers on the Square Root Spiral
The Ordered Distribution of Natural Numbers on the Square Root Spiral - Harry K. Hahn - Ludwig-Erhard-Str. 10 D-76275 Et Germanytlingen, Germany ------------------------------ mathematical analysis by - Kay Schoenberger - Humboldt-University Berlin ----------------------------- 20. June 2007 Abstract : Natural numbers divisible by the same prime factor lie on defined spiral graphs which are running through the “Square Root Spiral“ ( also named as “Spiral of Theodorus” or “Wurzel Spirale“ or “Einstein Spiral” ). Prime Numbers also clearly accumulate on such spiral graphs. And the square numbers 4, 9, 16, 25, 36 … form a highly three-symmetrical system of three spiral graphs, which divide the square-root-spiral into three equal areas. A mathematical analysis shows that these spiral graphs are defined by quadratic polynomials. The Square Root Spiral is a geometrical structure which is based on the three basic constants: 1, sqrt2 and π (pi) , and the continuous application of the Pythagorean Theorem of the right angled triangle. Fibonacci number sequences also play a part in the structure of the Square Root Spiral. Fibonacci Numbers divide the Square Root Spiral into areas and angle sectors with constant proportions. These proportions are linked to the “golden mean” ( golden section ), which behaves as a self-avoiding-walk- constant in the lattice-like structure of the square root spiral. Contents of the general section Page 1 Introduction to the Square Root Spiral 2 2 Mathematical description of the Square Root Spiral 4 3 The distribution -
(Spiral Curves) When Transition Curves Are Not Provided, Drivers Tend To
The Transition Curves (Spiral Curves) The transition curve (spiral) is a curve that has a varying radius. It is used on railroads and most modem highways. It has the following purposes: 1- Provide a gradual transition from the tangent (r=∞ )to a simple circular curve with radius R 2- Allows for gradual application of superelevation. Advantages of the spiral curves: 1- Provide a natural easy to follow path such that the lateral force increase and decrease gradually as the vehicle enters and leaves the circular curve 2- The length of the transition curve provides a suitable location for the superelevation runoff. 3- The spiral curve facilitate the transition in width where the travelled way is widened on circular curve. 4- The appearance of the highway is enhanced by the application of spiral curves. When transition curves are not provided, drivers tend to create their own transition curves by moving laterally within their travel lane and sometimes the adjoining lane, which is risky not only for them but also for other road users. Minimum length of spiral. Minimum length of spiral is based on consideration of driver comfort and shifts in the lateral position of vehicles. It is recommended that these two criteria be used together to determine the minimum length of spiral. Thus, the minimum spiral length can be computed as: Ls, min should be the larger of: , , . Ls,min = minimum length of spiral, m; Pmin = minimum lateral offset (shift) between the tangent and circular curve (0.20 m); R = radius of circular curve, m; V = design speed, km/h; C = maximum rate of change in lateral acceleration (1.2 m/s3) Maximum length of spiral. -
Topics in Complex Analytic Geometry
version: February 24, 2012 (revised and corrected) Topics in Complex Analytic Geometry by Janusz Adamus Lecture Notes PART II Department of Mathematics The University of Western Ontario c Copyright by Janusz Adamus (2007-2012) 2 Janusz Adamus Contents 1 Analytic tensor product and fibre product of analytic spaces 5 2 Rank and fibre dimension of analytic mappings 8 3 Vertical components and effective openness criterion 17 4 Flatness in complex analytic geometry 24 5 Auslander-type effective flatness criterion 31 This document was typeset using AMS-LATEX. Topics in Complex Analytic Geometry - Math 9607/9608 3 References [I] J. Adamus, Complex analytic geometry, Lecture notes Part I (2008). [A1] J. Adamus, Natural bound in Kwieci´nski'scriterion for flatness, Proc. Amer. Math. Soc. 130, No.11 (2002), 3165{3170. [A2] J. Adamus, Vertical components in fibre powers of analytic spaces, J. Algebra 272 (2004), no. 1, 394{403. [A3] J. Adamus, Vertical components and flatness of Nash mappings, J. Pure Appl. Algebra 193 (2004), 1{9. [A4] J. Adamus, Flatness testing and torsion freeness of analytic tensor powers, J. Algebra 289 (2005), no. 1, 148{160. [ABM1] J. Adamus, E. Bierstone, P. D. Milman, Uniform linear bound in Chevalley's lemma, Canad. J. Math. 60 (2008), no.4, 721{733. [ABM2] J. Adamus, E. Bierstone, P. D. Milman, Geometric Auslander criterion for flatness, to appear in Amer. J. Math. [ABM3] J. Adamus, E. Bierstone, P. D. Milman, Geometric Auslander criterion for openness of an algebraic morphism, preprint (arXiv:1006.1872v1). [Au] M. Auslander, Modules over unramified regular local rings, Illinois J. -
Phylogenetic Algebraic Geometry
PHYLOGENETIC ALGEBRAIC GEOMETRY NICHOLAS ERIKSSON, KRISTIAN RANESTAD, BERND STURMFELS, AND SETH SULLIVANT Abstract. Phylogenetic algebraic geometry is concerned with certain complex projec- tive algebraic varieties derived from finite trees. Real positive points on these varieties represent probabilistic models of evolution. For small trees, we recover classical geometric objects, such as toric and determinantal varieties and their secant varieties, but larger trees lead to new and largely unexplored territory. This paper gives a self-contained introduction to this subject and offers numerous open problems for algebraic geometers. 1. Introduction Our title is meant as a reference to the existing branch of mathematical biology which is known as phylogenetic combinatorics. By “phylogenetic algebraic geometry” we mean the study of algebraic varieties which represent statistical models of evolution. For general background reading on phylogenetics we recommend the books by Felsenstein [11] and Semple-Steel [21]. They provide an excellent introduction to evolutionary trees, from the perspectives of biology, computer science, statistics and mathematics. They also offer numerous references to relevant papers, in addition to the more recent ones listed below. Phylogenetic algebraic geometry furnishes a rich source of interesting varieties, includ- ing familiar ones such as toric varieties, secant varieties and determinantal varieties. But these are very special cases, and one quickly encounters a cornucopia of new varieties. The objective of this paper is to give an introduction to this subject area, aimed at students and researchers in algebraic geometry, and to suggest some concrete research problems. The basic object in a phylogenetic model is a tree T which is rooted and has n labeled leaves. -
Vector Bundles and Projective Modules
VECTOR BUNDLES AND PROJECTIVE MODULES BY RICHARD G. SWAN(i) Serre [9, §50] has shown that there is a one-to-one correspondence between algebraic vector bundles over an affine variety and finitely generated projective mo- dules over its coordinate ring. For some time, it has been assumed that a similar correspondence exists between topological vector bundles over a compact Haus- dorff space X and finitely generated projective modules over the ring of con- tinuous real-valued functions on X. A number of examples of projective modules have been given using this correspondence. However, no rigorous treatment of the correspondence seems to have been given. I will give such a treatment here and then give some of the examples which may be constructed in this way. 1. Preliminaries. Let K denote either the real numbers, complex numbers or quaternions. A X-vector bundle £ over a topological space X consists of a space F(£) (the total space), a continuous map p : E(Ç) -+ X (the projection) which is onto, and, on each fiber Fx(¡z)= p-1(x), the structure of a finite di- mensional vector space over K. These objects are required to satisfy the follow- ing condition: for each xeX, there is a neighborhood U of x, an integer n, and a homeomorphism <p:p-1(U)-> U x K" such that on each fiber <b is a X-homo- morphism. The fibers u x Kn of U x K" are X-vector spaces in the obvious way. Note that I do not require n to be a constant. -
Classical Algebraic Geometry
CLASSICAL ALGEBRAIC GEOMETRY Daniel Plaumann Universität Konstanz Summer A brief inaccurate history of algebraic geometry - Projective geometry. Emergence of ’analytic’geometry with cartesian coordinates, as opposed to ’synthetic’(axiomatic) geometry in the style of Euclid. (Celebrities: Plücker, Hesse, Cayley) - Complex analytic geometry. Powerful new tools for the study of geo- metric problems over C.(Celebrities: Abel, Jacobi, Riemann) - Classical school. Perfected the use of existing tools without any ’dog- matic’approach. (Celebrities: Castelnuovo, Segre, Severi, M. Noether) - Algebraization. Development of modern algebraic foundations (’com- mutative ring theory’) for algebraic geometry. (Celebrities: Hilbert, E. Noether, Zariski) from Modern algebraic geometry. All-encompassing abstract frameworks (schemes, stacks), greatly widening the scope of algebraic geometry. (Celebrities: Weil, Serre, Grothendieck, Deligne, Mumford) from Computational algebraic geometry Symbolic computation and dis- crete methods, many new applications. (Celebrities: Buchberger) Literature Primary source [Ha] J. Harris, Algebraic Geometry: A first course. Springer GTM () Classical algebraic geometry [BCGB] M. C. Beltrametti, E. Carletti, D. Gallarati, G. Monti Bragadin. Lectures on Curves, Sur- faces and Projective Varieties. A classical view of algebraic geometry. EMS Textbooks (translated from Italian) () [Do] I. Dolgachev. Classical Algebraic Geometry. A modern view. Cambridge UP () Algorithmic algebraic geometry [CLO] D. Cox, J. Little, D. -
Projective Geometry Lecture Notes
Projective Geometry Lecture Notes Thomas Baird March 26, 2014 Contents 1 Introduction 2 2 Vector Spaces and Projective Spaces 4 2.1 Vector spaces and their duals . 4 2.1.1 Fields . 4 2.1.2 Vector spaces and subspaces . 5 2.1.3 Matrices . 7 2.1.4 Dual vector spaces . 7 2.2 Projective spaces and homogeneous coordinates . 8 2.2.1 Visualizing projective space . 8 2.2.2 Homogeneous coordinates . 13 2.3 Linear subspaces . 13 2.3.1 Two points determine a line . 14 2.3.2 Two planar lines intersect at a point . 14 2.4 Projective transformations and the Erlangen Program . 15 2.4.1 Erlangen Program . 16 2.4.2 Projective versus linear . 17 2.4.3 Examples of projective transformations . 18 2.4.4 Direct sums . 19 2.4.5 General position . 20 2.4.6 The Cross-Ratio . 22 2.5 Classical Theorems . 23 2.5.1 Desargues' Theorem . 23 2.5.2 Pappus' Theorem . 24 2.6 Duality . 26 3 Quadrics and Conics 28 3.1 Affine algebraic sets . 28 3.2 Projective algebraic sets . 30 3.3 Bilinear and quadratic forms . 31 3.3.1 Quadratic forms . 33 3.3.2 Change of basis . 33 1 3.3.3 Digression on the Hessian . 36 3.4 Quadrics and conics . 37 3.5 Parametrization of the conic . 40 3.5.1 Rational parametrization of the circle . 42 3.6 Polars . 44 3.7 Linear subspaces of quadrics and ruled surfaces . 46 3.8 Pencils of quadrics and degeneration . 47 4 Exterior Algebras 52 4.1 Multilinear algebra .