Volume 15, issue 1 (2015)

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ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
An Abel map to the compactified Picard scheme realizes Poincaré duality

Jesse Leo Kass and Kirsten Wickelgren

Algebraic & Geometric Topology 15 (2015) 319–369
Abstract

For a smooth algebraic curve X over a field, applying H1 to the Abel map X PicXX to the Picard scheme of X modulo its boundary realizes the Poincaré duality isomorphism

H1(X, ) H1(XX, (1))H c1(X, (1)).

We show the analogous statement for the Abel map XX Pic¯XX to the compactified Picard, or Jacobian, scheme, namely this map realizes the Poincaré duality isomorphism H1(XX, ) H1(X, (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 XX 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 π1 Pic0()H1(, (1)).

Keywords
Abel map, compactified Picard scheme, compactified Jacobian, Poincaré duality
Mathematical Subject Classification 2010
Primary: 14F35
Secondary: 14D20, 14F20
References
Publication
Received: 15 January 2014
Revised: 24 June 2014
Accepted: 9 July 2014
Published: 23 March 2015
Authors
Jesse Leo Kass
Department of Mathematics
University of South Carolina
1523 Greene Street
Columbia, SC 29208
USA
http://www2.iag.uni-hannover.de/~kass/
Kirsten Wickelgren
School of Mathematics
Georgia Institute of Technology
686 Cherry Street
Atlanta, GA 30332
USA
http://people.math.gatech.edu/~kwickelgren3/