Rational and nearly rational varieties

Cover......Page 1
About......Page 2
Cambridge Studies in Advanced Mathematics 92......Page 3
Rational and Nearly Rational Varieties......Page 4
0521832071......Page 5
Contents......Page 6
Introduction�������������������......Page 8
Description of the chapters����������������������������������......Page 9
Prerequisites��������������������......Page 11
Notation and basic conventions�������������������������������������......Page 12
1.1 Rational and unirational varieties......Page 14
1.2 Rational curves......Page 16
1.3 Quadric hypersurfaces......Page 20
1.4 Quadrics over finite fields......Page 22
1.5 Cubic hypersurfaces......Page 27
1.6 Further examples of rational varieties......Page 36
1.7 Numerical criteria for nonrationality......Page 38
2 Cubic surfaces�����������������������......Page 42
2.1 The Segre-Manin theorem for cubic surfaces......Page 43
2.2 Linear systems on surfaces......Page 44
2.3 The proofs of the theorems of Segre and Manin......Page 49
2.4 Computing the Picard number of cubic surfaces......Page 54
2.5 Birational self-maps of the plane......Page 56
3 Rational surfaces��������������������������......Page 67
3.1 Castelnuovo’s rationality criterion......Page 68
3.2 Minimal models of surfaces......Page 70
3.3 Rational surfaces over perfect fields......Page 75
3.4 Field of definition of a subvariety......Page 83
3.5 Del Pezzo surfaces......Page 88
4 Nonrationality via reduction modulo p����������������������������������������������......Page 100
4.1 Nonrational cyclic covers......Page 101
4.2 Construction of cyclic covers......Page 104
4.3 Differential forms in characteristic p......Page 108
4.4 Reduction to characteristic p......Page 115
4.5 Matsusaka’s theorem and Abhyankar’s lemma......Page 118
4.6 Relative Canonical modules and Jacobian ideals......Page 121
4.7 Explicit examples (by J. Rosenberg)......Page 125
5 The Noether-Fano method for proving nonrationality�����������������������������������������������������������......Page 129
5.1 The Noether-Fano method......Page 130
5.2 Numerical consequences of maximal centers......Page 136
5.3 Birationally rigid Fano varieties......Page 141
5.4 Quartic threefolds......Page 151
6 Singularities of pairs�������������������������������......Page 156
6.1 Discrepancies......Page 157
6.2 Canonical and log canonical pairs......Page 162
6.3 Computing discrepancies......Page 164
6.4 Inversion of adjunction......Page 167
6.5 The log canonical threshold of a plane curve singularity......Page 171
6.6 Zero-dimensional maximal centers on threefolds......Page 182
6.7 Appendix: proof of the connectedness theorem......Page 184
7.1 Exercises in Chapter 1......Page 189
7.2 Exercises in Chapter 2......Page 201
7.3 Exercises in Chapter 3......Page 207
7.4 Exercises in Chapter 4......Page 218
7.5 Exercises in Chapter 5......Page 222
7.6 Exercises in Chapter 6......Page 224
References�����������������......Page 235
Index������������......Page 239