DOCSLIB.ORG

Collineation Groups of Projective. Planes and Spaces

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Collineation Groups of Projective. Planes and Spaces 1. COLLINEATION GROUPS OF PROJECTIVE. PLANES AND SPACES ......193221==1.1===11111 by FREDERICK CHARLES PIPER A Thesis submitted for the degree of Doctor of Philosophy in the University of London. IMPERIAL COLLEGE November, 1964 2 ABSTRACT We begin this thesis by answering the following questions. If EP is a finite projective plane and if-frs -i a collineation group of c3 such that - (i) FT- contains some elations of QP (ii)a line of P is the axis of elations inl-F- for more than one centre (iii)-rT - does not fix a point or line of then (a) what can be said about -FT ? (b) what configurations may be formed by the centres and the axes of these elations? We then use our answers to prove that if every flag of q) is fixed by an elation in-1T, then R is desarguesian, and either (i)11- contains the little projective group of @ , or, (ii) 3 6 the order of (3 is four andiT A or s6° In § 5 we show that if every point of is the centre of a homology in "TT and if no point or line of P is fixed by -TT- then (P is desarguesian, and Tr- contains the little projective group of CP . It is then shown that if the order of CP is not a square, then the same conclusions hold if we assume that every point is the centre of a perspectivity in IT Finally, in § 6, some of the results about elations in planes are extended to spaces of higher dimension. We prove the following; "Let Hk be a finite projective space of odd order and of 3 dimension k, whore k>21 and letTIbe a collineation group of Hi with the following properties contains some elations of Hk (ii) a hyparplane of Hk is the axis of elations in Tr for more than_ one centre (iii) no subspace of Hk is left fixed byll— thenTr contains the little projective group of Hk.H Also submitted as part of the thesis is a published note about elations in finite projective planes. 4. ACKNOWLEDGMEN'IS While preparing this Thesis I was helped by talks with Dr. P. Dembowski, Dr. D. H. Hughes and Dr. P. J. Higgins. I wish to thank them for their assistance, and to thank Dr. Higgins for acting as my supervisor while Dr. Wagner waa in America. Special thanks are due to my supervisor, Dr. A. Wagner, for the many hours spent in discussions, and for his endless supply of ideas, encouragement and criticisms. 5 CONTENTS 1. Definitions and basic concepts 00000 00 .. 00001000 6. 2. Intr oduction .. 0 0 00 000 o00 trao0c000000.00600 a0000 15. 3. Preliminary results ............. 000000000 00000• 22. 4. On elations of finite projective planes . 0 27. 5. Perspectivities of finite projective planes 55. 6. Elation of finite projective spaces of odd order 65. 7. References •00•000000000 . 000000 00000000 000 00000. 72. 8. Notation and definition of 'connected set' 75. X4RATA (see pages 68 & 69) Lemma 22 should be read before Lemma 21. The first two lines of the proof of Lemma 23. must be deleted, and should be replaced by "Let P and R be any two centres. By lemma 22, P has an axis, Hki say, such that 14 Hic.a. Let Sk bei an axis of R, let Q be any other centre of Rk.-i and let e- PQ." The rest of the proof of lemma 21 is correct. 6. 1. DEFINITIONS AND BASIC CONCEPTS 1.1 Projective Planes We define a projective plane as a set 6), the elements of which we call points, together with. a system of subsets of 6', which we call lines, satisfying the following three axioms:- 1. Given any two distinct points, there exists a unique line containing these points. 2. Given any two distinct lines, there exists a unique point contained in both these lines, 3. There exist four points such that no three of them are contained in one line. Throughout this thesis we shall use the following notation. Gothic upper case letters, e.g. CD , will be used for planes, sets of points and sets of lines. Roman upper case letters will be used for points and Roman lower case letters will be used for lines. If P and Q are any two distinct points and if e is the unique line containing them, we shall write e = K. If t.„1") and m are any two distinct lines and if 0 is the unique point contained in both of them, we shall write 0 = _e Ara. If P is any point and if (iis any line, we shall express that fact that P is contained in e_ by writing PE-2. If P is not contained in Ci , we shall write P,i..e.or, alternatively, PE 6) Q. If a projective plane contains only a finite number of points 7. the plane is said to be finite (Finite planes were first studied by Veblen and Bussey NI). It is easily proved that, for any finite projective plane 6), there exists a positive integer n such that CO every line of U' contains n + 1 points and (Li.) every point of 6P is on n 1 lines. This number is called the order of the plane. Simple counting shows that a finite projective plane of order n contains 2%1.n-a points and ii+n+1 lines. Any subset,, of the points of 6) which satisfies the axioms of a projective plane is said to be a subplane of 6). If every /n/ point of tr is on at least one lino of P, then O'is called a square-root subplane. This name arises from the fact that if 6? is, a finite projective plane of order n and if Oi ls a square-root subplane of 9, then n is a square and the order of V is If Pis a finite projective plane of oven (odd) order n, then a sot of n+2, (11-1), points such that no three are collinear is called an oval. (Ovals have bean extensively studied by s°gre [20, ?3)., An incident point-line pair is called a flag. Mappings of projective planes on to themselves play an important part in the theory of projective planes. Of particularimportance are the collineations. We define a collineation of a projective plane Pas a one-to-one mapping of the points of eon to the points of(dand of the linos ofe on. to the lines of Fsuch that incidence is preserved i.e. such that a point is on a line if, and only if, the image point is on the image lino. The mapping which results from the carrying out of two successive collineations is again a collineation. Defining the product of two collinoations as the collineation obtained by 8. successively performing them, it is readily seen that the set of all collineations of a projective plane form a multiplicative group. We call this group the full collinoation groat of the plane. Any subset of collineations which form a subgroup of the full collineation group is called a collineation group. Throughout this thesis we shall use Greek upper case letters to denote groups, particularly collineation groups, and. Greek lower case letters to denote elements of groups, particularly collineations. The one exception being that we shall denote the identity element of any group by I. The image of a point P under a collineation e4. will be written as Pa._ . Consequently Q0 is the collineation obtained by first performing 0: and then . cx, If a collinoation leaves fixed every point of some line, A -say, then 0( is. called a perspectivity. Baer DJ has shown that, if oCS, of...must also fix all the lines through some point, P say, and that can fix no other points or lines. The line e. is called the axis of the perspectivity and the point P is called its centre. We express the fact that P is the centre of 0( and that is its axis by calling c3( a (P,e)-perspectivity. If P£'€-WO call the perspectivity an. elation, and if it is called a homology. Unless indications are given to the contrary, we shall use the term elation (homology) to mean an elation (homology) other than the identity. The collinoation group generated by all the Elation of a desargues.ian plane 9 is called the little Projective gro..i.pt of Perspectivities assume a major role in the theory of projective planes. Their importance is due to their connection with the 9. Desargues configuration; a connection which was systematically studied by Baer [43. Two triangles ABC, A/ BI C/are said to be in perspective from 0 if OA = 0/11 , OB = OB' and OC = OCR . They are said to be in perspective from e if e /1AB = e Apb," BC = e A 131 CI and ACA = -e, /A/ It has been shown by Ostrom D13 that every, finite projective plane contains a. pair of triangles which are in. perspective from both a point and a line. A projective plane 63 will be called (P e)-desarguesian if, for every two triangles in perspective from the fixed point P and such that two pairs of corresponding sides intersect on the, fixed line 'e./.1 the intersection of the third pair of corresponding sides also lies on. e./. A projective plane will be said to be IL' t)- transitive if, for the fixed point P and the fixod line e-and any pair of points A, B where .A.,Bt Q, AfP, B P, PA=PB, there exists a (P, )-perspectivity 01C;Ith A etC =B.
Load more

Recommended publications
  • 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.
    [Show full text]
  • Robot Vision: Projective Geometry
    Robot Vision: Projective Geometry Ass.Prof. Friedrich Fraundorfer SS 2018 1 Learning goals . Understand homogeneous coordinates . Understand points, line, plane parameters and interpret them geometrically . Understand point, line, plane interactions geometrically . Analytical calculations with lines, points and planes . Understand the difference between Euclidean and projective space . Understand the properties of parallel lines and planes in projective space . Understand the concept of the line and plane at infinity 2 Outline . 1D projective geometry . 2D projective geometry ▫ Homogeneous coordinates ▫ Points, Lines ▫ Duality . 3D projective geometry ▫ Points, Lines, Planes ▫ Duality ▫ Plane at infinity 3 Literature . Multiple View Geometry in Computer Vision. Richard Hartley and Andrew Zisserman. Cambridge University Press, March 2004. Mundy, J.L. and Zisserman, A., Geometric Invariance in Computer Vision, Appendix: Projective Geometry for Machine Vision, MIT Press, Cambridge, MA, 1992 . Available online: www.cs.cmu.edu/~ph/869/papers/zisser-mundy.pdf 4 Motivation – Image formation [Source: Charles Gunn] 5 Motivation – Parallel lines [Source: Flickr] 6 Motivation – Epipolar constraint X world point epipolar plane x x’ x‘TEx=0 C T C’ R 7 Euclidean geometry vs. projective geometry Definitions: . Geometry is the teaching of points, lines, planes and their relationships and properties (angles) . Geometries are defined based on invariances (what is changing if you transform a configuration of points, lines etc.) . Geometric transformations
    [Show full text]
  • Collineations in Perspective
    Collineations in Perspective Now that we have a decent grasp of one-dimensional projectivities, we move on to their two di- mensional analogs. Although they are more complicated, in a sense, they may be easier to grasp because of the many applications to perspective drawing. Speaking of, let's return to the triangle on the window and its shadow in its full form instead of only looking at one line. Perspective Collineation In one dimension, a perspectivity is a bijective mapping from a line to a line through a point. In two dimensions, a perspective collineation is a bijective mapping from a plane to a plane through a point. To illustrate, consider the triangle on the window plane and its shadow on the ground plane as in Figure 1. We can see that every point on the triangle on the window maps to exactly one point on the shadow, but the collineation is from the entire window plane to the entire ground plane. We understand the window plane to extend infinitely in all directions (even going through the ground), the ground also extends infinitely in all directions (we will assume that the earth is flat here), and we map every point on the window to a point on the ground. Looking at Figure 2, we see that the lamp analogy breaks down when we consider all lines through O. Although it makes sense for the base of the triangle on the window mapped to its shadow on 1 the ground (A to A0 and B to B0), what do we make of the mapping C to C0, or D to D0? C is on the window plane, underground, while C0 is on the ground.
    [Show full text]
  • Finite Projective Geometries 243
    FINITE PROJECTÎVEGEOMETRIES* BY OSWALD VEBLEN and W. H. BUSSEY By means of such a generalized conception of geometry as is inevitably suggested by the recent and wide-spread researches in the foundations of that science, there is given in § 1 a definition of a class of tactical configurations which includes many well known configurations as well as many new ones. In § 2 there is developed a method for the construction of these configurations which is proved to furnish all configurations that satisfy the definition. In §§ 4-8 the configurations are shown to have a geometrical theory identical in most of its general theorems with ordinary projective geometry and thus to afford a treatment of finite linear group theory analogous to the ordinary theory of collineations. In § 9 reference is made to other definitions of some of the configurations included in the class defined in § 1. § 1. Synthetic definition. By a finite projective geometry is meant a set of elements which, for sugges- tiveness, are called points, subject to the following five conditions : I. The set contains a finite number ( > 2 ) of points. It contains subsets called lines, each of which contains at least three points. II. If A and B are distinct points, there is one and only one line that contains A and B. HI. If A, B, C are non-collinear points and if a line I contains a point D of the line AB and a point E of the line BC, but does not contain A, B, or C, then the line I contains a point F of the line CA (Fig.
    [Show full text]
  • Estimating Projective Transformation Matrix (Collineation, Homography)
    Estimating Projective Transformation Matrix (Collineation, Homography) Zhengyou Zhang Microsoft Research One Microsoft Way, Redmond, WA 98052, USA E-mail: [email protected] November 1993; Updated May 29, 2010 Microsoft Research Techical Report MSR-TR-2010-63 Note: The original version of this report was written in November 1993 while I was at INRIA. It was circulated among very few people, and never published. I am now publishing it as a tech report, adding Section 7, with the hope that it could be useful to more people. Abstract In many applications, one is required to estimate the projective transformation between two sets of points, which is also known as collineation or homography. This report presents a number of techniques for this purpose. Contents 1 Introduction 2 2 Method 1: Five-correspondences case 3 3 Method 2: Compute P together with the scalar factors 3 4 Method 3: Compute P only (a batch approach) 5 5 Method 4: Compute P only (an iterative approach) 6 6 Method 5: Compute P through normalization 7 7 Method 6: Maximum Likelihood Estimation 8 1 1 Introduction Projective Transformation is a concept used in projective geometry to describe how a set of geometric objects maps to another set of geometric objects in projective space. The basic intuition behind projective space is to add extra points (points at infinity) to Euclidean space, and the geometric transformation allows to move those extra points to traditional points, and vice versa. Homogeneous coordinates are used in projective space much as Cartesian coordinates are used in Euclidean space. A point in two dimensions is described by a 3D vector.
    [Show full text]
  • A Modular Compactification of the General Linear Group
    Documenta Math. 553 A Modular Compactification of the General Linear Group Ivan Kausz Received: June 2, 2000 Communicated by Peter Schneider Abstract. We define a certain compactifiction of the general linear group and give a modular description for its points with values in arbi- trary schemes. This is a first step in the construction of a higher rank generalization of Gieseker's degeneration of moduli spaces of vector bundles over a curve. We show that our compactification has simi- lar properties as the \wonderful compactification” of algebraic groups of adjoint type as studied by de Concini and Procesi. As a byprod- uct we obtain a modular description of the points of the wonderful compactification of PGln. 1991 Mathematics Subject Classification: 14H60 14M15 20G Keywords and Phrases: moduli of vector bundles on curves, modular compactification, general linear group 1. Introduction In this paper we give a modular description of a certain compactification KGln of the general linear group Gln. The variety KGln is constructed as follows: First one embeds Gln in the obvious way in the projective space which contains the affine space of n × n matrices as a standard open set. Then one succes- sively blows up the closed subschemes defined by the vanishing of the r × r subminors (1 ≤ r ≤ n), along with the intersection of these subschemes with the hyperplane at infinity. We were led to the problem of finding a modular description of KGln in the course of our research on the degeneration of moduli spaces of vector bundles. Let me explain in some detail the relevance of compactifications of Gln in this context.
    [Show full text]
  • Certain Properties of Orbits Under Collineation Groups*
    JOURNAL OF COMBINATORIAL THEORY 2, 546-570 (1967) Certain Properties of Orbits under Collineation Groups* DAVID A. FOULSER AND REUBEN SANDLER University of lllinois at Chicago Circle, Chicago, lllinois Communicated by Gian-Carlo Rota l. INTRODUCTION" In this paper, the authors examine some of the combinatorial proper- ties of orbits of projective planes under collineation groups. Dembowski, in [5], has studied in some detail the relations which can hold between a point orbit ~ and a line orbit ~. Our approach here is to associate with the set s of all lines incident with two or more points of ~, and to look at the numerical relationships which occur. In Section 2, the basic concepts are defined, and some necessary condi- tions are derived which must hold if a collineation group with certain properties can exist. Section 3 discusses the insufficiency of these condi- tions and gives some results yielding sufficiency in some special cases; in addition, further necessary conditions are developed. In Section 4, a more detailed analysis is made of the nature of the interaction between and ~(~) under special conditions on the collineation group G. These special conditions are analogous to requiring that G be Abelian -- class (A)--or of prime order -- class (I-I)--and very detailed information is available under these hypotheses. Finally, Section 5 contains a num- ber of examples illustrating the principles discussed in the earlier sections. Standard notation for permutation groups is used throughout this paper. See for example [11]. * This work was partially supported by NSF contract No. GP 5276. 546 CERTAIN PROPERTIES OF ORBITS UNDER COLLINEATION GROUPS 547 2.
    [Show full text]
  • Euclidean Versus Projective Geometry
    Projective Geometry Projective Geometry Euclidean versus Projective Geometry n Euclidean geometry describes shapes “as they are” – Properties of objects that are unchanged by rigid motions » Lengths » Angles » Parallelism n Projective geometry describes objects “as they appear” – Lengths, angles, parallelism become “distorted” when we look at objects – Mathematical model for how images of the 3D world are formed. Projective Geometry Overview n Tools of algebraic geometry n Informal description of projective geometry in a plane n Descriptions of lines and points n Points at infinity and line at infinity n Projective transformations, projectivity matrix n Example of application n Special projectivities: affine transforms, similarities, Euclidean transforms n Cross-ratio invariance for points, lines, planes Projective Geometry Tools of Algebraic Geometry 1 n Plane passing through origin and perpendicular to vector n = (a,b,c) is locus of points x = ( x 1 , x 2 , x 3 ) such that n · x = 0 => a x1 + b x2 + c x3 = 0 n Plane through origin is completely defined by (a,b,c) x3 x = (x1, x2 , x3 ) x2 O x1 n = (a,b,c) Projective Geometry Tools of Algebraic Geometry 2 n A vector parallel to intersection of 2 planes ( a , b , c ) and (a',b',c') is obtained by cross-product (a'',b'',c'') = (a,b,c)´(a',b',c') (a'',b'',c'') O (a,b,c) (a',b',c') Projective Geometry Tools of Algebraic Geometry 3 n Plane passing through two points x and x’ is defined by (a,b,c) = x´ x' x = (x1, x2 , x3 ) x'= (x1 ', x2 ', x3 ') O (a,b,c) Projective Geometry Projective Geometry
    [Show full text]
  • ON the STRUCTURE of COLLINEATION GROUPS of FINITE PROJECTIVE PLANES by WILLIAM M
    ON THE STRUCTURE OF COLLINEATION GROUPS OF FINITE PROJECTIVE PLANES By WILLIAM M. KANTORf [Received 31 December 1973—Revised 24 September 1974] 1. Introduction The study of a collineation group F of a finite projective plane SP splits into two parts: the determination first of the abstract structure of F, and then of the manner in which F acts on 3P. If F contains no perspectivities, almost nothing general is known about it. We shall study both questions in the case where F is generated by involutory perspectivities. To avoid unstructured situations, F will be assumed to contain at least two such perspectivities having different centres or axes. The determination of the structure of F is then not difficult if the order n of 2P is even (see § 8, Remark 1). Consequently, we shall concentrate on the case of odd n. The main reason for the difficulties occurring for odd n is that the permutation representation of F on the set of centres (or axes) of involutory perspectivi- ties has no nice group-theoretic properties; this is the exact opposite of the situation occurring for even n. As usual, Z(T) and O(F) will denote the centre and largest normal subgroup of odd order of F; av denotes the conjugacy class of a in F; and Cr(A) and <A> are the centralizer of, and subgroup generated by, the subset A. THEOREM A. Let ^ be a finite projective plane of odd order n, and F a collineation group of & generated by involutory homologies and 0(T). Assume that F contains commuting involutory homologies having different axes, and that there is no involutory homology a for which oO{T) e Z(F/O{T)).
    [Show full text]
  • Frobenius Collineations in Finite Projective Planes
    Frobenius Collineations in Finite Projective Planes Johannes Ueberberg 1 Introduction Given a finite field F = GF (qn)oforderqn it is well-known that the map f : F ! F , f : x 7! xq is a field automorphism of F of order n, called the Frobenius automor- phism.IfV is an n-dimensional vector space over the finite field GF (q), then V can be considered as the vector space of the field GF (qn)overGF (q). Therefore the Frobenius automorphism induces a linear map over GF (q) R : V ! V R : x 7! xq of order n on V . It follows that R induces a projective collineation ' on the (n − 1)- dimensional projective space PG(n−1;q). We call ' and any projective collineation conjugate to ' a Frobenius collineation. In the present paper we shall study the case n = 3, that is, the Frobenius collineations of the projective plane PG(2;q). Let P = PG(2;q2). Then every Singer cycle σ (see Section 3) of P defines a partition P(σ)ofthepointsetofP into pairwise disjoint Baer subplanes. These partitions are called linear Baer partitions or, equivalently, Singer Baer partitions [17]. If % is a Frobenius collineation of P , then we define E% to be the set of Baer subplanes of P fixed by %. It turns out that for q ≡ 2mod3wehavejP(σ) \E%j2 2 f0; 1; 3g with jP(σ) \E%j =3ifandonlyif% 2 NG(<σ>), where G = PGL3(q ) (see 3.5). Received by the editors March 1996. Communicated by J. Thas. 1991 Mathematics Subject Classification : 51E20, 51E24. Key words and phrases : diagram geometry, Baer partitions, Frobenius collineations.
    [Show full text]
  • Parallelisms of Quadric Sets
    Innovations in Incidence Geometry ACADEMIA PRESS Volume 12 (2011), Pages 21–34 ISSN 1781-6475 Parallelisms of quadric sets William E. Cherowitzo Norman L. Johnson Abstract In this article, it is shown that every flock of a hyperbolic quadric H and every flock of a quadratic cone C in PG(3,K), for K a field, is in a transitive parallelism of H or C, respectively. Furthermore, it is shown it is possible to have parallelisms of quadratic cones by maximal partial flocks. The theory of parallelisms of quadratic cones is generalized to analogous results for parallelisms of α-cones. Keywords: flocks, flokki, parallelisms, hyperbolic quadric, elliptic quadric, quadratic cone, α-cone MSC 2000: 51E20, 51A15 1 Introduction Let K be a field and consider a flock of a quadratic cone, an elliptic quadric or a hyperbolic quadric in PG(3,K). This article considers whether there are “parallelisms” of these quadric sets, which in each case means a set of mutually disjoint flocks, whose union is a complete cover of the set of conics of plane intersections of the quadric set in question. Consider first a “hyperbolic parallelism” as a union of mutually disjoint hyper- bolic flocks, whose union is the set of all conics that are sections of a hyperbolic quadric. When K is isomorphic to GF(q), there are (q4 1)/(q 1) planes in PG(3,q), − − and there are 2(q +1) lines that lie in the hyperbolic quadric. Each of these lines lies in q + 1 planes, none of which can be associated with a hyperbolic flock and each plane that contains a line of a ruling also contains a line of the opposite ruling.
    [Show full text]
  • Foundations of Projective Geometry
    Foundations of Projective Geometry Robin Hartshorne 1967 ii Preface These notes arose from a one-semester course in the foundations of projective geometry, given at Harvard in the fall term of 1966–1967. We have approached the subject simultaneously from two different directions. In the purely synthetic treatment, we start from axioms and build the abstract theory from there. For example, we have included the synthetic proof of the fundamental theorem for projectivities on a line, using Pappus’ Axiom. On the other hand we have the real projective plane as a model, and use methods of Euclidean geometry or analytic geometry to see what is true in that case. These two approaches are carried along independently, until the first is specialized by the introduction of more axioms, and the second is generalized by working over an arbitrary field or division ring, to the point where they coincide in Chapter 7, with the introduction of coordinates in an abstract projective plane. Throughout the course there is special emphasis on the various groups of transformations which arise in projective geometry. Thus the reader is intro- duced to group theory in a practical context. We do not assume any previous knowledge of algebra, but do recommend a reading assignment in abstract group theory, such as [4]. There is a small list of problems at the end of the notes, which should be taken in regular doses along with the text. There is also a small bibliography, mentioning various works referred to in the preparation of these notes. However, I am most indebted to Oscar Zariski, who taught me the same course eleven years ago.
    [Show full text]