An Abel map to the compactified Picard scheme realizes Poincaré duality

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

$$H_1\left(X,\mathbb{Z}∕\ell \right)\to H^1\left(X∕\partial X,\mathbb{Z}∕\ell \left(1\right)\right)\cong H_c^1\left(X,\mathbb{Z}∕\ell \left(1\right)\right).$$

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 $H_1\left(X∕\partial X,\mathbb{Z}∕\ell \right)\to H^1\left(X,\mathbb{Z}∕\ell \left(1\right)\right)$. 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 ${\pi }_1^{\ell }{Pic}^0\left(-\right)\cong H^1\left(-,{\mathbb{Z}}_{\ell }\left(1\right)\right).$