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.