Lectures on Curves, Surfaces and Projective Varieties: A Classical View of Algebraic GeometryEuropean Mathematical Society, 2009 - 491 páginas 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. |
Índice
Preface | 1 |
Segre Varieties | 11 |
2 | 12 |
Bibliography | 31 |
Geometric Properties of Algebraic Varieties | 51 |
Rudiments of Elimination Theory | 83 |
Hypersurfaces in Projective Space | 106 |
7 | 139 |
Linear Systems | 167 |
Algebraic Curves | 199 |
Linear Series on Algebraic Curves | 238 |
8 | 274 |
Cremona Transformations | 292 |
Rational Surfaces | 343 |
433 | 454 |
43 | 469 |
Otras ediciones - Ver todo
Lectures on Curves, Surfaces and Projective Varieties: A Classical View of ... Mauro Beltrametti,Ettore Carletti,Giacomo Monti Bragadin,Dionisio Gallarati No hay ninguna vista previa disponible - 2009 |
Términos y frases comunes
adjoints of order affine algebraic correspondence algebraic set base points base variety belonging Bézout's theorem bijective birational isomorphism canonical divisors canonical series coincides common complete series conics consider contained coordinates Cremona transformation cubic curve of order defined dimension distinct double point equation equivalent example finite number fixed points genus given Hence homaloidal system homogeneous polynomials hyperplane sections hypersurface of order ideal intersection multiplicity linear series linear space linear system locus matrix morphism multiple points non-singular obtains open subset osculating parameters passing plane curve polar projective image projective space Proof Proposition quadratic transformation quadric Q quartic rational map respect ruled surface s-fold point series g simple point subspace suppose surface F t₁ tangent cone tangent hyperplane tangent plane Theorem topology Veronese surface vertex x₁ y₁ zero дхо