ALGEBRAISCHE GEOMETRIE

Transcription

1 ERGEBNISSE DER MATHEMATIK UND IHRER GRENZGEBIETE UNTER MITWIRKUNG DER SCHRIFTLEITUNG DES "ZENTRALBLATT FOR MATHEMATIK" HERAUSGEGEBEN VON L.V.AHLFORS R.BAER R.COURANT ].L.DOOB S.EILENBERG P. R. HALMOS. T. NAKAYAMA H. RADEMACHER F.K. SCHMIDT B. SEGRE E. SPERNER ====== NEUE FOLGE HEFT 13 ====== REIHE: ALGEBRAISCHE GEOMETRIE BESORGT VON B. SEGRE S PRINGER-VERLAG BERLIN GOTTINGEN. HEIDELBERG 1957

2 SOME PROPERTIES OF DIFFERENTIABLE VARIETIES AND TRANSFORMATIONS WITH SPECIAL REFERENCE TO THE ANALYTIC AND ALGEBRAIC CASES BY BENIAMINO SEGRE S PRINGER-VERLAG BERLIN GOTTINGEN HEIDELBERG 1957

3 ALLE RECHTE, INSBESONDERE DAS DER UBERSETZUNG IN FREMDE SPRACHEN, VORBEHALTEN OHNE AUSDRUCKLICHE GENEHMIGUNG DES VERLAGES 1ST ES AUCH NICHT GESTATTET, DIESES BUCH ODER TEILE DARAUS AUF PHOTOMECHANISCHEM WEGE (PHOTOKOPIE, MIKROKOPIE) ZU VERVIELFALTIGEN ISBN ISBN (ebook) DOI / BY SPRINGER-VERLAG OHG. BERLIN' GOTTINGEN HEIDELBERG 1957 BRUHLSCHE UNIVERSITATSDRUCKEREI GIESSEN

4 Preface The present volume contains, together with numerous additions and extensions, the course of lectures which I gave at Pavia (26 September till 5 October 1955) by invitation of the «Centro Internazionale Matematico Estivo». The treatment has the character of a monograph, and presents various novel features, both in form and in substance; these are indicated in the notes which will be found at the beginning and end of each chapter. Of the nine parts into which the work is divided, the first four are essentially differential in character, the next three deal with algebraic geometry, while the last two are concerned with certain aspects of the theory of differential equations and of correspondences between topological varieties. A glance at the index will suffice to give a more exact idea of the range and variety of the contents, whose chief characteristic is that of establishing suggestive and sometimes unforeseen relations between apparently diverse subjects (e. g. differential geometry in the small and also in the large, algebraic geometry, function theory, topology, etc.); prominence is given throughout to the geometrical viewpoint, and tedious calculations are as far as possible avoided. The exposition has been planned so that it can be followed without much difficulty even by readers who have no special knowledge of the subjects treated. Although, for reasons of space, various proofs are here only sketched, nevertheless the essence of the methods is always made clear, while, on the other hand, at the end of each chapter there are sufficient references to enable the reader to pursue matters further. However, in certain cases cognate results have been omitted for the sake of brevity; for the same reason, the Bibliography does not claim to be complete. I have had the good fortune to have received valuable assistance from Dr. DAVID KIRBY in preparing the English version of the text, from Dr. EDOARDO VESENTINI in typing and revising the typescript, and from Prof. LEoNARD ROTH in reading the proofs; to all these I tender hearty and grateful thanks. In conclusion, I wish to express to the Springer-Verlag my appreciation of the superlatively good printing and production of this book. Roma, 22. II BENIAMINO SEGRE

5 Contents Part One. Differential Invariants of Point and Dual Transformations 1. Local metrical study of point transformations 2. Some topologico-differential invariants 3 3. Projective construction of the above invariants 5 4. Local metrical study of the dual transformations 7 5. Calculation of the first order differential invariants just considered 9 6. Some particular transformations. Relations between densities I I 7. The curvature of hypersurfaces and of pfaffian forms 12 Historical Notes and Bibliography Part Two. Local Properties of Analytic Transformations at their United Points Coefficients of dilatation and residues of transformations in the analytic field Transfer to the Riemann variety Formal changes of coordinates Formal reduction to the canonical form for the arithmetically general transformations The case of arithmetically special transformations Criteria of convergence for the reduction procedure in the general case Iteration and permutability of analytic transformations On the united points of cyclic analytic transformations Arithmetically general transformations not representable linearly 31 Historical Notes and Bibliography Part Three. Invariants of Contact and of Osculation. The Concept of Cross-ratio in Differential Geometry Projective invariants of two curves having the same osculating spaces at a point A notable metric case An important extension Projective invariants of contact of differential elements of any dimension Two applications On certain varieties generated by quadrics The notion of cross-ratio on certain surfaces Applications to various branches of differential geometry Some extensions Historical Notes and Bibliography Part Four. Principal and Projective Curves of a Surface, and Some Applications Some results of projective-differential geometry The definition and main properties of the principal and projective curves Further properties of the above curves The use of the Laplace invariants and of the infinitesimal invariants Some classes of surfaces on which the concept of cross-ratio is particularly simple

6 Contents Point correspondences which conserve the projective curves 32. Point correspondences which preservc the principal lines 33. On the plane cone curves of a surface Historical Notes and Bibliography Part Five. Some Differential Properties in the Large of Algebraic Curves, their Intersections, and Self-correspondences The residues of correspondences on curves, and a topological invariant of intersection of two curves on a surface which contains two privileged pencils of curves A complcment of the correspondence principle on algebraic curves A geometric characterization of abelian integrals and their residues The first applications The equation of Jacobi, and some consequences The relation of Reiss, and some extensions Further algebro-differential properties 86 Historical Notes and Bibliography. 88 Part Six. Extensions to Algebraic Varieties Generalizations of the equations of Jacobi Generalization of the relation of Reiss The residue of an analytic transformation at a simple united point Some important particular cases Relations between residues at the same point The total residue of correspondences of valency zero on algebraic varieties The residues at isolated united points with arbitrary multiplicities Extensions to algebraic correspondences of arbitrary valency Applications to algebraic correspondences of a projective space into itself Historical Notes and Bibliography Part Seven. Veronese Varieties and Modules of Algebraic Forms n-regular points of differentiable varieties Some special properties of n-regular points of differentiable varieties On the freedom of hypersurfaces having assigned multiplicities at a set of points On the effective dimension of certain linear systems of hypersurfaces Two relations of Lasker concerning modules of hypersurfaces Some important criteria for a hypersurface to belong to a given module Some properties of the osculating spaces at the points of a Veronese variety v~n) The ambients of certain subvarieties of v~n) The isolated multiple intersections of d primals on v~n) The regular multiple intersections on v~n) A special property of the space associated with an isolated intersection on v~n) in the simple case On a theorem of Torelli and some complements 130 Historical Notes and Bibliography 131 Part Eight. Linear Partial Differential Equations Preliminary observations The reduction of differential equations to a canonical form Remarks on the solution of the differential equations The construction of the conditions of integrability

7 8 Contents 66. The conditions of compatibility for a system of linear partial differential equations in one unknown The analytic case where the characteristic hypersurfaces intersect regularly An extension to the non-analytic case Some remarks on sets of linear partial differential equations in several unknowns The solution of a system of homogeneous equations The resolving system associated with a general set of m differential equations in m unknowns Historical Notes and Bibliography Part Nine. Correspondences between Topological Varieties Products of topological varieties Correspondences and relations Inverse correspondences Homologous correspondences.. ISS 76. Topological invariants of correspondences between topological varieties Arithmetic and algebraic invariants Geometric invariants i -correspondences on topological varieties Semiregular correspondences and their products Characteristic integers of a semi-regular correspondence Involutory elementary s-correspondences Algebraic and skew-algebraic involutory transformations An extension of Zeuthen's formula to the topological domain One-valued elementary correspondences Correspondences represented by differentiable varieties 169 Historical Notes and Bibliography 171 Bibliography. 171 Author Index. 179 Analytic Index 181