An Abel map to the compactified Picard scheme realizes Poincaré duality
© 2015 Mathematical Sciences Publishers. All rights reserved. For a smooth algebraic curve X over a field, applying H1 to the Abel map X→PicX∕∂X to the Picard scheme of X modulo its boundary realizes the Poincaré duality isomorphism H1(X,Z∕ℓ)→H1(X∕∂X,Z∕ℓ(1))≅H1c(X,Z∕ℓ(1)). We show the analogous statement for the Abel map X/∂X→Pic X/∂X to the compactified Picard, or Jacobian, scheme, namely this map realizes the Poincaré duality isomorphism H1(X∕∂X,Z∕ℓ)→H1(X,Z∕ℓ(1)). In particular, H1 of this Abel map is an isomorphism. In proving this result, we prove some results about Pic that are of independent interest. The singular curve X∕∂X has a unique singularity that is an ordinary fold point, and we describe the compactified Picard scheme of such a curve up to universal homeomorphism using a presentation scheme. We construct a Mayer–Vietoris sequence for certain pushouts of schemes, and an isomorphism of functors πℓ1Pic0(−)≅H1(−,Zℓ(1)).
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