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Towards Birational Classification of Algebraic Varieties

Towards Birational Classification of Algebraic Varieties TOWARDS BIRATIONAL CLASSIFICATION OF ALGEBRAIC VARIETIES P. M. H. WILSON Introduction The aim of this survey article is to describe the present state of classification theory for complex algebraic varieties in view of the substantial progress made by Mori, Kawamata and others during the last few years. It is hoped that the account will be understandable to the general mathematician with a background knowledge of algebraic geometry, although such a reader may wish to skip some of the details in the later sections. Recall that any irreducible variety V defined over the complex numbers has a tne function field C(V), field of rational functions on V. We say that two algebraic varieties are birationally equivalent if there is a birational map between them; this happens if and only if their function fields are isomorphic over C. CLASSICAL PROBLEM. Classify irreducible complex varieties up to birational equivalence; stated in algebraic terms the problem is to classify finitely generated extension fields of C up to C-isomorphism. We aim to describe varieties up to birational equivalence by means of'invariants', that is, quantities associated with a variety which are invariant under birational equivalence. The most obvious of these is the dimension, which by http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

Towards Birational Classification of Algebraic Varieties

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Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/19.1.1
Publisher site
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Abstract

TOWARDS BIRATIONAL CLASSIFICATION OF ALGEBRAIC VARIETIES P. M. H. WILSON Introduction The aim of this survey article is to describe the present state of classification theory for complex algebraic varieties in view of the substantial progress made by Mori, Kawamata and others during the last few years. It is hoped that the account will be understandable to the general mathematician with a background knowledge of algebraic geometry, although such a reader may wish to skip some of the details in the later sections. Recall that any irreducible variety V defined over the complex numbers has a tne function field C(V), field of rational functions on V. We say that two algebraic varieties are birationally equivalent if there is a birational map between them; this happens if and only if their function fields are isomorphic over C. CLASSICAL PROBLEM. Classify irreducible complex varieties up to birational equivalence; stated in algebraic terms the problem is to classify finitely generated extension fields of C up to C-isomorphism. We aim to describe varieties up to birational equivalence by means of'invariants', that is, quantities associated with a variety which are invariant under birational equivalence. The most obvious of these is the dimension, which by

Journal

Bulletin of the London Mathematical Society – Wiley

Published: Jan 1, 1987

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