Lectures on Curves, Surfaces and Projective Varieties: A Classical View of Algebraic Geometry (Ems Textbooks in Mathematics)

This book offers a wide-ranging introduction to algebraic geometry along classical lines. It consists of lectures on topics in classical algebraic geometry, including the basic properties of projective algebraic varieties, linear systems of hypersurfaces, algebraic curves (with special emphasis on rational curves), linear series on algebraic curves, Cremona transformations, rational surfaces, and notable examples of special varieties like the Segre, Grassmann, and Veronese varieties. An integral part and special feature of the presentation is the inclusion of many exercises, not easy to find in the literature and almost all with complete solutions. The text is aimed at students in the last two years of an undergraduate program in mathematics. It contains some rather advanced topics suitable for specialized courses at the advanced undergraduate or beginning graduate level, as well as interesting topics for a senior thesis. The prerequisites have been deliberately limited to basic elements of projective geometry and abstract algebra. Thus, for example, some knowledge of the geometry of subspaces and properties of fields is assumed. The book will be welcomed by teachers and students of algebraic geometry who are seeking a clear and panoramic path leading from the basic facts about linear subspaces, conics and quadrics to a systematic discussion of classical algebraic varieties and the tools needed to study them. The text provides a solid foundation for approaching more advanced and abstract literature. Preface......Page 7 Contents......Page 13 1 Prerequisites......Page 17 Review of topology......Page 28 The correspondences V and I......Page 31 Morphisms......Page 37 Rational maps......Page 40 Projective algebraic sets......Page 46 Rational maps and birational equivalence......Page 51 Complements and exercises......Page 59 Tangent space, singularities and dimension......Page 67 Independence of polynomials. Essential parameters......Page 78 Dimension of a projective variety......Page 84 Order of a projective variety, tangent cone and multiplicity......Page 88 Resultant of two polynomials......Page 99 Bézout's theorem for plane curves......Page 105 More on intersection multiplicity......Page 106 Elimination of several variables......Page 115 Bézout's theorem......Page 118 Generalities on hypersurfaces......Page 122 Multiple points of a hypersurface......Page 124 Algebraic envelopes......Page 131 Polarity with respect to a hypersurface......Page 135 Quadrics in projective space......Page 142 Complements on polars......Page 150 Plane curves......Page 156 Surfaces in P3......Page 165 Linear systems of hypersurfaces......Page 182 Hypersurfaces of a linear system that satisfy given conditions......Page 184 Base points of a linear system......Page 186 Jacobian loci......Page 193 Simple, composite, and reducible linear systems......Page 198 Rational mappings......Page 201 Projections and Veronese varieties......Page 205 Blow-ups......Page 208 Generalities......Page 213 The genus of an algebraic curve......Page 217 Curves on a quadric......Page 228 Rational curves......Page 234 Exercises on rational curves......Page 241 8 Linear Series on Algebraic Curves......Page 254 Divisors on an algebraic curve with ordinary singularities......Page 255 Linear series......Page 262 Linear equivalence......Page 264 Projective image of linear series......Page 267 Special linear series......Page 272 Adjoints and the Riemann–Roch theorem......Page 277 Properties of the canonical series and canonical curves......Page 286 Some results on algebraic correspondences between two curves......Page 290 Some remarks regarding moduli......Page 293 Complements and exercises......Page 300 Quadratic transformations between planes......Page 308 Resolution of the singularities of a plane algebraic curve......Page 313 Cremona transformations between planes......Page 323 Cremona transformations between projective spaces of dimension 3......Page 333 Exercises......Page 338 Planar representation of rational surfaces......Page 356 Linearly normal surfaces and their projections......Page 365 Surfaces of minimal order......Page 370 The conics of a plane as points of P5 and the Veronese surface......Page 376 Complements and exercises......Page 380 The product of two projective lines......Page 402 Segre morphism and Segre varieties......Page 405 Segre product of varieties......Page 408 Examples and exercises......Page 411 The lines of P3 as points of a quadric in P5......Page 415 Complexes of lines in P3......Page 419 Congruences of lines in P3......Page 423 Ruled surfaces in P3......Page 424 Grassmann coordinates and Grassmann varieties......Page 430 Further properties of G(1,n) and applications......Page 438 Miscellaneous exercises......Page 449 Further problems......Page 470 Exercises on linear series on curves......Page 473 Bibliography......Page 483 Index......Page 491 "This book offers a wide-ranging introduction to algebraic geometry along classical lines. It consists of lectures on topics in classical algebraic geometry, including the basic properties of projective algebraic varieties, linear systems of hypersurfaces, algebraic curves (with special emphasis on rational curves), linear series on algebraic curves, Cremona transformations, rational surfaces, and notable examples of special varieties like the Segre, Grassmann, and Veronese varieties. An integral part and special feature of the presentation is the inclusion of many exercises, not easy to find in the literature and almost all with complete solutions. The text is aimed at students of the last two years of an undergraduate program in mathematics. It contains some rather advanced topics suitable for specialized courses on the advanced undergraduate or beginning graduate level, as well as interesting topics for a senior thesis. The prerequisites have been deliberately limited to basic elements of projective geometry and abstract algebra. Thus, for example, some knowledge of the geometry of subspaces and properties of fields is assumed. The book will be welcomed by teachers and students of algebraic geometry who are seeking a clear and panoramic path leading from the basic facts about linear subspaces, conics and quadrics to a systematic discussion of classical algebraic varieties and the tools needed to study them. The text provides a solid foundation for approaching more advanced and abstract literature."--Page 4 de la couverture Prerequisites -- Algebraic Sets, Morphisms, And Rational Maps -- Geometric Properties Of Algebraic Varieties -- Rudiments Of Elimination Theory -- Hypersurfaces In Projective Space -- Linear Systems -- Algebraic Curves -- Linear Series On Algebraic Curves -- Cremona Transformations -- Rational Surfaces -- Segre Varieties -- Grassmann Varieties -- Supplementary Exercises. Mauro C. Beltrametti ... [et Al.] ; Translated From The Italian By Francis Sullivan. Original Title: Letture Su Curve, Superficie E Varietà Proiettive Speciali : Un'introduzione Alla Geometria Algebrica, Published By Bollati Boringhieri, Torino, C2003. Includes Bibliographical References (p. [467]-473) And Index.