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An Abel map to the compactified Picard scheme realizes Poincaré duality

Publication ,  Journal Article
Kass, JL; Wickelgren, K
Published in: Algebraic and Geometric Topology
March 23, 2015

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 H1(X,Z∕ℓ)→H1(X∕∂X,Z∕ℓ(1))≅H1c(X,Z∕ℓ(1)). 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 πℓ1Pic0(−)≅H1(−,Zℓ(1)).

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Published In

Algebraic and Geometric Topology

DOI

EISSN

1472-2739

ISSN

1472-2747

Publication Date

March 23, 2015

Volume

15

Issue

1

Start / End Page

319 / 369

Related Subject Headings

  • General Mathematics
  • 4904 Pure mathematics
  • 0101 Pure Mathematics
 

Citation

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Kass, J. L., & Wickelgren, K. (2015). An Abel map to the compactified Picard scheme realizes Poincaré duality. Algebraic and Geometric Topology, 15(1), 319–369. https://doi.org/10.2140/agt.2015.15.319
Kass, J. L., and K. Wickelgren. “An Abel map to the compactified Picard scheme realizes Poincaré duality.” Algebraic and Geometric Topology 15, no. 1 (March 23, 2015): 319–69. https://doi.org/10.2140/agt.2015.15.319.
Kass JL, Wickelgren K. An Abel map to the compactified Picard scheme realizes Poincaré duality. Algebraic and Geometric Topology. 2015 Mar 23;15(1):319–69.
Kass, J. L., and K. Wickelgren. “An Abel map to the compactified Picard scheme realizes Poincaré duality.” Algebraic and Geometric Topology, vol. 15, no. 1, Mar. 2015, pp. 319–69. Scopus, doi:10.2140/agt.2015.15.319.
Kass JL, Wickelgren K. An Abel map to the compactified Picard scheme realizes Poincaré duality. Algebraic and Geometric Topology. 2015 Mar 23;15(1):319–369.

Published In

Algebraic and Geometric Topology

DOI

EISSN

1472-2739

ISSN

1472-2747

Publication Date

March 23, 2015

Volume

15

Issue

1

Start / End Page

319 / 369

Related Subject Headings

  • General Mathematics
  • 4904 Pure mathematics
  • 0101 Pure Mathematics