## Abelian varieties |

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### Contents

Algebraic Groups | 1 |

The field of definition of a group variety | 13 |

Chapter II | 19 |

Copyright | |

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A X B abelian subvariety abelian variety defined adic Albanese variety algebraic closure algebraic groups algebraically equivalent arbitrary variety assume automorphism birational isomorphism canonical map Chapter Chow's theorem codimension complete and non-singular component concludes the proof contained Corollary cycles of dimension defined over k(t degree denote element endomorphism EndQ equal exists a function F-VII6 Th factor group field of definition field of rationality finite formula group variety hence hyperplane section independent generic points induced integer intersection theory inverse image isogeny Jacobian kernel law of composition Lemma Let f linear equivalence class linear system linearly equivalent Math morphism multiplicity non-singular in codimension numerical equivalence obtain Picard variety point of G Pontrjagin product positive divisor primary extension Proposition purely inseparable extension rational map rational point regular extension shows simple point subvariety Suppose surjective translation trivial Tt(A U X V uniquely determined universal mapping property V X W