An Abel map to the compactified Picard scheme realizes Poincar\'e duality

TL;DR

This paper demonstrates that the Abel map to the compactified Picard scheme of a singular curve realizes Poincaré duality, extending classical results to singular curves and providing new insights into their Picard schemes.

Contribution

It establishes that the Abel map to the compactified Picard scheme realizes Poincaré duality for singular curves, and develops new tools for analyzing Picard schemes of such curves.

Findings

The Abel map induces an isomorphism in homology and cohomology groups.

The compactified Picard scheme of a singular curve with an ordinary fold point is characterized up to universal homeomorphism.

A Mayer-Vietoris sequence for certain scheme push-outs is constructed.

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 Poincar\'e 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 Poincar\'e 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 \pi_1^{ell} Pic^0(-) = H^1(-,Z_ell(1)).