Mathieu Gagne

Autoduality of the Compactified Jacobian

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 Mumfords, 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.

Abel maps and presentation schemes

We sharpen the two main tools used to treat the compactified Jacobian of a singular curve: Abel maps and presentation schemes. First we prove a smoothness theorem for bigraded Abel maps. Second we study the two complementary filtrations provided by the images of certain Abel maps and certain presentation schemes. Third we study a lifting of the Abel map of bidegree (m, 1) to the corresponding presentation scheme. Fourth we prove that, if a curve is blown up at a double point, then the corresponding presentation scheme is a 1-bundle. Finally, using Abel maps of bidegree (m, 1), we characterize the curves having double points at worst.