© 2002 by London Mathematical Society
© The London Mathematical Society
Autoduality of the Compactified Jacobian
Instituto de Matemática Pura e Aplicada, Estrada D Castorina 110, 22460-320 Rio de Janeiro RJ, Brazil, esteves{at}impa.br
EMC Corporation 171 South Street, Hopkinton, MA 01748, USA, mgagne{at}emc.com
Room 2-278, Department of Mathematics, Massachusetts Institute of Technology 77 Massachusetts Avenue, Cambridge, MA 02139-4307, USA kleiman{at}math.mit.edu
Received 2 January 2001.
The following autoduality theorem is proved for an integral projective curve C in any characteristic. Given an invertible sheaf L of degree 1, form the corresponding Abel map AL:C
, which maps C into its compactified Jacobian, and form its pullback map 
, which carries the connected component of 0 in the Picard scheme back to the Jacobian. If C has, at worst, points of multiplicity 2, then 
is an isomorphism, and forming it commutes with specializing C.
Much of the work in the paper is valid, more generally, for a family of curves with, at worst, points of embedding dimension 2. In this case, the determinant of cohomology is used to construct a right inverse to . Then a scheme-theoretic version of the theorem of the cube is proved, generalizing Mumford's, and it is used to prove that is independent of the choice of L. Finally, the autoduality theorem is proved. The presentation scheme is used to achieve an induction on the difference between the arithmetic and geometric genera; here, special properties of points of multiplicity 2 are used.