  The purpose of this article is to show that the Castelnuovo theory for
abelian varieties, developed by G. Pareschi and M. Popa, can be
infinitesimalized. More precisely, we prove that an irreducible principally
polarized abelian variety has a finite scheme in extremal position, in the
sense of Castelnuovo theory for abelian varieties, if, and only if, it is a
Jacobian and the scheme is contained in a unique Abel-Jacobi curve.
