#### Volume 15, issue 1 (2015)

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An Abel map to the compactified Picard scheme realizes Poincaré duality

### Jesse Leo Kass and Kirsten Wickelgren

Algebraic & Geometric Topology 15 (2015) 319–369
##### Abstract

For a smooth algebraic curve $X$ over a field, applying ${H}_{1}$ to the Abel map $X\to PicX∕\partial X$ to the Picard scheme of $X$ modulo its boundary realizes the Poincaré duality isomorphism

${H}_{1}\left(X,ℤ∕\ell \right)\to {H}^{1}\left(X∕\partial X,ℤ∕\ell \left(1\right)\right)\cong {H}_{c}^{1}\left(X,ℤ∕\ell \left(1\right)\right).$

We show the analogous statement for the Abel map $X∕\partial X\to \overline{Pic}X∕\partial X$ to the compactified Picard, or Jacobian, scheme, namely this map realizes the Poincaré duality isomorphism ${H}_{1}\left(X∕\partial X,ℤ∕\ell \right)\to {H}^{1}\left(X,ℤ∕\ell \left(1\right)\right)$. In particular, ${H}_{1}$ of this Abel map is an isomorphism.

In proving this result, we prove some results about $\overline{Pic}$ 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 pushouts of schemes, and an isomorphism of functors ${\pi }_{1}^{\ell }{Pic}^{0}\left(-\right)\cong {H}^{1}\left(-,{ℤ}_{\ell }\left(1\right)\right).$

##### Keywords
Abel map, compactified Picard scheme, compactified Jacobian, Poincaré duality
##### Mathematical Subject Classification 2010
Primary: 14F35
Secondary: 14D20, 14F20