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Compactifying the relative Jacobian over families of reduced curves

Author: Eduardo Esteves
Journal: Trans. Amer. Math. Soc. 353 (2001), 3045-3095
MSC (2000): Primary 14H40, 14H60; Secondary 14D20
Published electronically: January 18, 2001
MathSciNet review: 1828599
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Abstract: We construct natural relative compactifications for the relative Jacobian over a family $X/S$ of reduced curves. In contrast with all the available compactifications so far, ours admit a Poincaré sheaf after an étale base change. Our method consists of studying the étale sheaf $F$ of simple, torsion-free, rank-1 sheaves on $X/S$, and showing that certain open subsheaves of $F$ have the completeness property. Strictly speaking, the functor $F$ is only representable by an algebraic space, but we show that $F$ is representable by a scheme after an étale base change. Finally, we use theta functions originating from vector bundles to compare our new compactifications with the available ones.

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Additional Information

Eduardo Esteves
Affiliation: Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina 110, 22460-320 Rio de Janeiro RJ, Brazil

Received by editor(s): December 15, 1997
Received by editor(s) in revised form: May 2, 2000
Published electronically: January 18, 2001
Additional Notes: Research supported by an MIT Japan Program Starr fellowship, by PRONEX, Convênio 41/96/0883/00 and CNPq, Proc. 300004/95-8.
Article copyright: © Copyright 2001 American Mathematical Society

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