RESUMEN
We were led to the question of embedding curves in
smooth surfaces by our study [I], done in collaboration with A. Iarrobino, of the irreducibility
of the compactified jacobian of a complete integral
curve. We had proved that irreducibility holds if the curve lies on a smooth
surface. Previously, D’Souza
[D] had proved that irreducibility holds is the curve is smooth except for
simple nodes and simple cusps. Since the local embedding dimensions at simple
nodes and cusps are 2, our result includes D’Souza’s. Moreover, we needed Theorem (1) below to
show how sharp our irreducibility result
is; we used it to help construct an irreducible complete intersection in P³
that is smooth except for one singular point, whose compactified jacobian
is irreducible.