AUTODUALITY OF THE COMPACTIFIED JACOBIAN

Abstract

The following autoduality theorem is proved for an integral projective curve $C$ in any characteristic. Given an invertible sheaf ${\cal L}$ of degree 1, form the corresponding Abel map $A_{\cal L}:C\longrightarrow \bar{J}$ , which maps $C$ into its compactified Jacobian, and form its pullback map $A^{\ast}_{\cal L}:{\rm Pic}^0_{\bar{J}}\longrightarrow J$ , 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 $A^{\ast}_{\cal L}$ 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 $A^{\ast}_{\cal L}$ . Then a scheme-theoretic version of the theorem of the cube is proved, generalizing Mumford's, and it is used to prove that $A^{\ast}_{\cal L}$ is independent of the choice of ${\cal L}$ . 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.