Castelnuovo function, zero-dimensional schemes and singular plane curves

We study families V of curves in P-2(C) of degree cl having exactly r singu lar points of given topological or analytic types. We derive new sufficient conditions for V to be T-smooth (smooth of the expected dimension), respec tively to be irreducible. For T-smoothness these conditions involve new inv ariants of curve singularities and are conjectured to be asymptotically pro per, that is, optimal up to a constant factor; for curves with nodes and cu sps these conditions are indeed optimal up to linear terms in d. To obtain the results, we study the Castelnuovo function, prove the irreducibility of the Hilbert scheme of zero-dimensional schemes associated to a cluster of infinitely near points of the singularities and deduce new vanishing theore ms for ideal sheaves of sere-dimensional schemes in P-2. Moreover, we give a series of examples of cuspidal curves where the family V is reducible, bu t where pi(1)(P-2\C) coincides (and is abelian) for all C is an element of V.