Dolgachev I.V. Classical algebraic geometry: a modern view. - Cambridge; New York: Cambridge University Press, 2012. - xii, 639 p.: ill. - Ref.: p.593-619 - Sub. ind.: p.620-639. - ISBN 978-1-107-01765-8  Место хранения:   013 | Институт математики СО РАН | Новосибирск | Библиотека

Оглавление / Contents

Preface ........................................................ xi 1 Polarity ..................................................... 1 1.1 Polar hypersurfaces ..................................... 1 1.2 The dual hypersurface .................................. 28 1.3 Polar s-hedra .......................................... 35 1.4 Dual homogeneous forms ................................. 48 1.5 First examples ......................................... 60 Exercises ................................................... 64 Historical notes ............................................ 66 2 Conies and quadric surfaces ................................. 69 2.1 Self-polar triangles ................................... 69 2.2 Poncelet relation ...................................... 81 2.3 Quadric surfaces ....................................... 91 Exercises .................................................. 108 Historical notes ........................................... 111 3 Plane cubics ............................................... 114 3.1 Equations ............................................. 114 3.2 Polars of a plane cubic ............................... 124 3.3 Projective generation of cubic curves ................. 133 3.4 Invariant theory of plane cubics ...................... 136 Exercises .................................................. 141 Historical notes ........................................... 143 4 Determinantal equations .................................... 146 4.1 Plane curves .......................................... 146 4.2 Determinantal equations for hypersurfaces ............. 160 Exercises .................................................. 184 Historical notes ........................................... 186 5 Theta characteristics ...................................... 188 5.1 Odd and even theta characteristics .................... 188 5.2 Hyperelliptic curves .................................. 192 5.3 Theta functions ....................................... 197 5.4 Odd theta characteristics ............................. 204 5.5 Scorza correspondence ................................. 212 Exercises .................................................. 224 Historical notes ........................................... 224 6 Plane quartics ............................................. 226 6.1 Bitangents ............................................ 226 6.2 Determinant equations of a plane quartic .............. 235 6.3 Even theta characteristics ............................ 243 6.4 Invariant theory of plane quartics .................... 265 6.5 Automorphisms of plane quartic curves ................. 266 Exercises .................................................. 276 Historical notes ........................................... 278 7 Cremona transformations .................................... 280 7.1 Homaloidal linear systems ............................. 280 7.2 First examples ........................................ 294 7.3 Planar Cremona transformations ........................ 303 7.4 Elementary transformations ............................ 320 7.5 Noether's Factorization Theorem ....................... 329 Exercises .................................................. 342 Historical notes ........................................... 344 8 del Pezzo surfaces ......................................... 347 8.1 First properties ...................................... 347 8.2 The Etf-lattice ....................................... 358 8.3 Anticanonical models .................................. 379 8.4 del Pezzo surfaces of degree ≥ 6 ...................... 386 8.5 del Pezzo surfaces of degree 5 ........................ 389 8.6 Quartic del Pezzo surfaces ............................ 396 8.7 del Pezzo surfaces of degree 2 ........................ 405 8.8 del Pezzo surfaces of degree 1 ........................ 411 Exercises .................................................. 422 Historical notes ........................................... 423 9 Cubic surfaces ............................................. 426 9.1 Lines on a nonsingular cubic surface .................. 426 9.2 Singularities ......................................... 443 9.3 Determinantal equations ............................... 449 9.4 Representations as sums of cubes ...................... 459 9.5 Automorphisms of cubic surfaces ....................... 483 Exercises .................................................. 502 Historical notes .............................................. 504