## Rational and Nearly Rational VarietiesThe 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. |

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### Contents

Introduction | 1 |

Quadrics over finite fields | 14 |

Further examples of rational varieties | 29 |

The proofs of the theorems of Segre and Manin | 42 |

Rational surfaces | 60 |

Field of definition of a subvariety | 76 |

Nonratiimulity via reduction modulo p | 93 |

The NoetherFano method for proving nonrationality | 122 |

Singularities of pairs | 149 |

Solutions to exercises | 186 |

228 | |

### Common terms and phrases

affine algehraically closed field anti-canonical arithmetic genus assume base locus base point free birational map birational morphism birational transform birationally blowing blowup codimension coefficients compute consider contained coordinates critical points cubic hypersurface cubic surface cyclic cover defined Del Pezzo surface denote differential forms dimension discrepancy equation example exceptional divisor Exercise Fano varieties finite Galois geometrically rational given ground field hence homogeneous implies intersection number irreducible components isomorphic Lemma line bundle linear system linearly equivalent log canonical threshold log resolution maximal center mobile linear system module monomials Newton polygon pair tX Pezzo surface Picard group Picard number plane conic polynomial project ive proof of Theorem Proposition prove pull-back quadratic quadric quartic threefold rational curve rational map rational varieties resuh Sarkisov degree self-intersection sheaf singular point smooth cubic surface smooth projective smooth surface smooth variety SOLUTION subscheme subvariety system F unirational vanishing weighted projective space