An Abel map to the compactified Picard scheme realizes Poincaré duality

Journal Article

© 2015 Mathematical Sciences Publishers. All rights reserved. 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)).

Full Text

Duke Authors

Cited Authors

  • Kass, JL; Wickelgren, K

Published Date

  • March 23, 2015

Published In

Volume / Issue

  • 15 / 1

Start / End Page

  • 319 - 369

Electronic International Standard Serial Number (EISSN)

  • 1472-2739

International Standard Serial Number (ISSN)

  • 1472-2747

Digital Object Identifier (DOI)

  • 10.2140/agt.2015.15.319

Citation Source

  • Scopus