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.