elliptic - Annali di Matematica 183, 317331 (2004) Digital...

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Digital Object Identifer (DOI) 10.1007/s10231-003-0094-0 Annali di Matematica 183, 317–331 (2004) Igor V. Dolgachev On certain families of elliptic curves in projective space ? To the memory of Fabio Bardelli Received: September 18, 2002; in fnal Form: November 10, 2002 Published online: ±ebruary 16, 2004 – Springer-Verlag 2004 Mathematics Subject ClassiFcation (2000). 14E07, 14H25, 14N20 1. Introduction Let El ( n ; p 1 ,... , p m ) be the Family oF elliptic curves oF degree n + 1in P n containing a fxed set oF m distinct points p 1 p m . In modern language, El ( n ; p 1 p m ) is a Zariski-open subset oF the fbre oF the evaluation map ev : M 1 , m ( P n , n + 1 ) ( P n ) m ,whe re M 1 , m ( P n , n + 1 ) is the space oF regular maps oF elliptic curves equipped with an ordered set oF m points to a curve oF degree n + P n . Its general fbre, iF not empty, is an irreducible variety oF dimension ( n + 1 ) 2 m ( n 1 ) . The largest possible m For which the map ev is dominant is equal to 9 ( n = 2 , 5 ), 8 ( n = 3 , 4 ), n + 3 ( n 6 ) . In the last case we show that the general fbre is isomorphic to an open subset oF a complete intersection oF n 2 diagonal quadrics in P n + 2 . In particular, birationally, it is a ±ano variety iF n 6, a Calabi–Yau iF n = 7, and oF general type iF n 8. The group G n = ( Z / 2 Z ) n + 2 acts naturally in P n + 2 by multiplying the projective coordinates with ± 1. The cor- responding action oF a subgroup oF index 2 oF G n is induced by a certain group oF Cremona transFormations in P n which we will describe explicitly. There are three cases when El ( n ; p 1 p m ) is oF expected dimension 0. They are ( n , m ) = ( 2 , 9 ), ( 3 , 8 ), ( 5 , 9 ) . It is well known that in the frst two cases El ( n ; p 1 p m ) consists oF one point. Less known is the Fact that the same is true in the case ( 5 , 9 ) . D. Babbage [1] attributes this result to T. G. Room. Apparently it was proven much earlier by A. Coble [2]. We reproduce Coble’s prooF in the paper. This result implies the existence oF a rational elliptic fbration f : P 5 −→ El ( 5 , p 1 p 8 ) which is an analog oF the well-known rational elliptic fbrations P 2 P 1 = El ( 2 ; p 1 p 8 ) and P 3 P 2 = El ( 3 ; p 1 p 7 ) defned by the pencil oF plane cubics through 8 points and by the net oF quadrics through 7 I.V. Dolgachev: Department oF Mathematics, University oF Michigan, Ann Arbor, MI 48109, USA, e-mail: [email protected] ? Research partially supported by NS± grant DMS 990780
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318 I.V. Dolgachev points, respectively. We show that its locus of points of indeterminacy is a certain 3-fold, a Weddle variety studied intensively by Coble [4], [5], [6]. I am grateful to D. Eisenbud, S. Mukai and R. Vakil for valuable discussions on the topic of this paper. 2. Association 2.1. We start with a reminder of the classical theory of association of Fnite sets of points. We follow the modern exposition of this theory given in [8]. Let Z be a Gorenstein scheme of dimension 0 over a Feld k , L be an invertible sheaf on Z and V H 0 ( Z , L ) be a linear system. The duality pairing H 0 ( Z , L ) × H 0 ( Z Z L 1 ) H 0 ( Z Z ) trace −−→ k allows one to deFne the subspace V H 0 ( Z Z L 1 ) . The pair ( V Z L 1 ) is called the Gale transform of ( V , L ) .
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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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elliptic - Annali di Matematica 183, 317331 (2004) Digital...

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