design - ABSTRACT CONFIGURATIONS IN ALGEBRAIC GEOMETRY I...

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Unformatted text preview: ABSTRACT CONFIGURATIONS IN ALGEBRAIC GEOMETRY I. DOLGACHEV To the memory of Andrei Tyurin Abstract. An abstract ( v k ,b r )-configuration is a pair of finite sets of cardinalities v and b with a relation on the product of the sets such that each element of the first set is related to the same number k of elements from the second set and, conversely, each element of the second set is related to the same number r of el- ements in the first set. An example of an abstract configuration is a finite geometry. In this paper we discuss some examples of abstract configurations and, in particular finite geometries, which one encounters in algebraic geometry. CONTENTS 1. Introduction 2. Configurations, designs and finite geometries 3. Configurations in algebraic geometry 4. Modular configurations 5. The Ceva configurations 6. v 3-configurations 7. The Reye (12 4 , 16 3 )-configuration 8. v v − 1-configurations 9. The Cremona-Richmond 15 3-configuration 10. The Kummer configurations 11. A symmetric realization of P 2 ( F q ) 1. Introduction In this paper we discuss some examples of abstract configurations and, in particular finite geometries, which one encounters in algebraic geometry. The Fano Conference makes it very appropriate because of the known contribution of Gino Fano to finite geometry. In his first published paper [18] he gave a first synthetical definition of the projective plane over an arbitrary field. In [18] introduces his famous Fano’s Postulate. It asserts that the diagonals of a complete quadrangle do not intersect at one point. Since this does not hold for the projective plane over a finite field of two elements (the Fano plane), the postulate 1 2 I. DOLGACHEV allows one to exclude the case of characteristic 2. Almost forty years later he returned to the subject of finite projective planes in [19]. An abstract ( v k ,b r )- configuration is a pair of finite sets of cardinal- ities v and b with a relation on the product of the sets such that each element of the first set is related to the same number k of elements from the second set and, conversely, each element of the second set is related to the same number r of elements in the first set. A geometric realization of an abstract configuration is a realization of each set as a set of linear (or affine, or projective) subspaces of certain dimension with a certain incidence relation. Any abstract configuration admits a geometric realization in a projective space of sufficient large dimen- sion over an infinite field. The existence of a geometric realization in a given space over a given field, for example, by real points and lines in the plane, is a very difficult problem. A regular configuration is an abstract configuration which admits a group of automorphisms which acts transitively on the sets A and B . Many regular configurations arise in algebraic geometry, where the sets A and B are represented by sub- varieties of an algebraic variety with an appropriate incidence relation....
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This note was uploaded on 02/24/2012 for the course MATH 285 taught by Professor Igordolgachev during the Fall '04 term at University of Michigan-Dearborn.

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design - ABSTRACT CONFIGURATIONS IN ALGEBRAIC GEOMETRY I...

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