目录 PREFACE BOOK III GENERAL THEORY OF ALGEBRAIC VARIETIES IN PROJECTIVE SPACE CHAAPTER X: ALGEBRAIC VARIETIES 1.Introduction 2.Reducible and irreducible varieties 3.Generic points of an irreducible variety 4.Generic members of systems of k-spaces 5.The dimension of an algebraic variety 6.The Cayley form of an algebraic variety 7.Properties of the Cayley form 8.Further properties of the Cayley form 9.The order of an algebraic variety; parametrisation 10.Some algebraic lemmas 11.Absolutely and relatively irreducible varieties 12.Some properties of relatively irreducible varieties 13.Sections of an absolutely irreducible variety 14.Tangent spaces and simple points CHAPTER XI: ALGEBRAIC CORRESPONDENCES 1.Varieties in r-way projective space 2.Segres representation of r-way projective space 3.Two-way algebraic correspondences 4.The Principle of Counting Constants 5.A special correspondence 6.Systems of algebraic varieties and related correspondences 7.Normal problems 8.Multiplicative varieties 9.A criterion for unit multiplicity 10.Simple points CHAPTER XII: INTERSECTION THEORY 1.Introduction 2.The degeneration of an irreducible variety in Sn 3.The product and cross.j oin of two irreducible varieties in Sn 4.The intersection of two irreducible varieties in Sn 5.Intersection theory in Sn 6.The intersection of irreducible varieties on a Vn in Sn 7.Intersection theory on a non-singular Vn 8.Intersections of systems of varieties 9.Equivalence on an algebraic variety 10.Virtual varieties 11.Theory of the base BOOK IV QUADRICS AND GRASSMANN VARIETIES CHAPTER XIII: QUADRICS 1.Definitions and elementary properties 2.Quadric primals in Sn 3.Polar theory of quadrics 4.Linear spaces on a quadric, I 5.Linear spaces on a quadric, II 6.The subvarieties of a quadric 7.Stereographic projection 8.The projective transfor- mations of a quadric into itself 9.The elementary divisors of orthogonal matrices 10.Pairs of quadrics 11.The intersection of two quadrics in Sn CHAPTER XIV: GRASSMANN VARIETIES 1.Grassmann varieties 2.Schubert varieties 3.Equations of a Schubert variety 4.Intersections of Schubert varieties: point-set properties 5.The basis theorem 6.The intersection formulae 7.Applications to enumerative geometry 8.Varieties of dimension (n-d)(d+1)--I onΩ(d, n) 9.Postulation formulae for BIBLIOGRAPHICAL NOTES BIBLIOGRAPHY INDEX
精彩内容 This Volume gives an account of the principal methods used in developing a theory of algebraic varieties in space of n dimensions. Applications of these methods are also given to some of the more important varieties which occur in projective geometry. It wasorigina113 our intention to include an account of the arithmetic theory of varieties， and of the foundations of birational geometry， but it has turned out to be more convenient to reserve these topics for a third volume. The theory of algebraic varieties developed in this volume is therefore mainly a theory of varieties in projective space. In writing this volume we have been faced with two problems： the difficult question of what must go in and what should be left out， and the problem of the degree of generality to be aimed at. As our objective has been to give an account of the modern algebraic methods available to geometers， we have not sought generality for its own sake. There is still enough to be done in the realm of classical geometry to give these methods all the scope that could be desired， and had it been possible to confine ourselves to the classical case of geometry over the field of complex numbers， we should have been content to do so. But in order to put the classical methods on a sound basis， using algebraic methods， it is necessary to consider geometry over more general fields than the field of complex numbers. However， if the ultimate object is to provide a sound algebraic basis for classical geometry， it is only necessary to consider fields without characteristic. Since geometry over any field without characteristic conforms to the general pattern of geometry over the field of complex numbers， we have developed the theory of algebraic varieties over any field without characteristic. Thus fields with finite characteristic are not used in this book.
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