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An Abel map to the compactified Picard scheme realizes Poincaré duality

Jesse Leo Kass and Kirsten Wickelgren

Algebraic & Geometric Topology 15 (2015) 319–369

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)).

Abel map, compactified Picard scheme, compactified Jacobian, Poincaré duality
Mathematical Subject Classification 2010
Primary: 14F35
Secondary: 14D20, 14F20
Received: 15 January 2014
Revised: 24 June 2014
Accepted: 9 July 2014
Published: 23 March 2015
Jesse Leo Kass
Department of Mathematics
University of South Carolina
1523 Greene Street
Columbia, SC 29208
Kirsten Wickelgren
School of Mathematics
Georgia Institute of Technology
686 Cherry Street
Atlanta, GA 30332