Compactifying the relative Jacobian over families of reduced curves

Esteves, Eduardo

doi:10.1090/S0002-9947-01-02746-5

Compactifying the relative Jacobian over families of reduced curves
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Trans. Amer. Math. Soc. 353 (2001), 3045-3095 Request permission

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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