An Abel map to the compactified Picard scheme realizes Poincaré duality
arXiv:1305.6330 · doi:10.2140/agt.2015.15.319
Abstract
For a smooth algebraic curve X over a field, applying H_1 to the Abel map X -> Pic (X/\partial X) to the Picard scheme of X modulo its boundary realizes the Poincaré duality isomorphism H_1(X, Z/ n) -> H^1(X/ \partial X, Z/n(1)) = H^1_c(X, Z/n(1)). We show the analogous statement for the Abel map X/\partial X -> Picbar (X/\partial X) to the compactified Picard, or Jacobian, scheme, namely this map realizes the Poincaré duality isomorphism H_1(X/ \partial X, Z/n) -> H^1(X, Z/n(1)). In particular, H_1 of this Abel map is an isomorphism. In proving this result, we prove some results about Picbar that are of independent interest. The singular curve X/\partial 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 push-outs of schemes, and an isomorphism of functors π_1^{ell} Pic^0(-) = H^1(-,Z_ell(1)).
arXiv comment50 pages. To appear in Algebraic and Geometric Topology. This version corrects an error in the argument showing that Example 4.1 is as claimed. (The example itself was correct.)