An Abel map to the compactified Picard scheme realizes Poincare duality

作者：Kass Jesse Leo^*; Wickelgren Kirsten

来源：Algebraic and Geometric Topology, 2015, 15(1): 319-369.

DOI：10.2140/agt.2015.15.319

For a smooth algebraic curve X over a field, applying H-1 to the Abel map X -> Pic X/partial derivative X to the Picard scheme of X modulo its boundary realizes the Poincare duality isomorphism
H-1 (X, Z/l) -> H-1(X/partial derivative X, Z/l(1)) congruent to H-c(1) (X, Z/l(1)).
We show the analogous statement for the Abel map X/partial derivative X -> (P) over bar ic X/partial derivative X to the compactified Picard, or Jacobian, scheme, namely this map realizes the Poincare duality isomorphism H-1 (X/partial derivative X, Z/l) -> H-1 (X, Z/l(1)). In particular, H-1 of this Abel map is an isomorphism.
In proving this result, we prove some results about (P) over bar ic that are of independent interest. The singular curve X/partial derivative 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 pi(l)(1) Pic0 (-) congruent to H-1 (-, Z(l)(1)).

• 出版日期2015