Rational and Nearly Rational VarietiesCambridge University Press, 22 avr. 2004 - 235 pages The most basic algebraic varieties are the projective spaces, and rational varieties are their closest relatives. In many applications where algebraic varieties appear in mathematics and the sciences, we see rational ones emerging as the most interesting examples. The authors have given an elementary treatment of rationality questions using a mix of classical and modern methods. Arising from a summer school course taught by János Kollár, this book develops the modern theory of rational and nearly rational varieties at a level that will particularly suit graduate students. There are numerous examples and exercises, all of which are accompanied by fully worked out solutions, that will make this book ideal as the basis of a graduate course. It will act as a valuable reference for researchers whilst helping graduate students to reach the point where they can begin to tackle contemporary research problems. |
Table des matières
Introduction | 1 |
Examples of rational varieties | 7 |
Cubic surfaces | 35 |
The proofs of the theorems of Segre and Manin | 42 |
Birational selfmaps of the plane | 49 |
Nonrationality via reduction modulo p | 93 |
The NoetherFano method for proving nonrationality | 122 |
20 | 133 |
31 | 142 |
Singularities of pairs | 149 |
35 | 164 |
42 | 170 |
47 | 176 |
3 | 182 |
| 228 | |
Expressions et termes fréquents
1-curves affine algebraically closed field arithmetic genus assume base locus base point free birational map birational morphism birational transform blowup C₁ codimension coefficients compute conic consider contained coordinates critical points cubic surface cyclic cover defined Del Pezzo surface denote dimension discrepancy equation exceptional divisor Exercise F₁ Fano varieties finite ground field hyperplane hypersurface intersection number inversion of adjunction invertible sheaf irreducible components isomorphism k-points Kollár Lemma line bundle linear system linearly equivalent log canonical threshold log resolution maximal center mobile linear system multiplicity multp Newton polygon Noether-Fano method nonrational normal variety P₁ pair Pezzo surface Picard group Picard number plane polynomial proof of Theorem Proposition 5.11 prove pull-back Q-divisor Q-linear combination quadric quartic threefold sheaf singular point smooth projective smooth variety special fiber subscheme subvariety Theorem 4.4 vanishing weighted projective space