An Abel map to the compactified Picard scheme realizes Poincaré duality

Kass, Jesse¹ ; Wickelgren, Kirsten²

¹Department of Mathematics, University of South Carolina, 1523 Greene Street, Columbia, SC 29208, USA
²School of Mathematics, Georgia Institute of Technology, 686 Cherry Street, Atlanta, GA 30332, USA

Algebraic and Geometric Topology, Tome 15 (2015) no. 1, pp. 319-369

Résumé

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

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 π1ℓ Pic0(−)≅H1(−, ℤℓ(1)).

Détail

DOI : 10.2140/agt.2015.15.319

Classification : 14F35, 14D20, 14F20
Keywords: Abel map, compactified Picard scheme, compactified Jacobian, Poincaré duality

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