Castelnuovo function, zero-dimensional schemes and singular plane curves

We study families V of curves in ℙ² (ℂ) of degree d having exactly r singular points of given topological or analytic types. We derive new sufficient conditions for V to be T-smooth (smooth of the expected dimension), respectively to be irreducible. For T-smoothness these conditions involve new invariants of curve singularities and are conjectured to be asymptotically proper, that is, optimal up to a constant factor; for curves with nodes and cusps 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 theorems for ideal sheaves of zero-dimensional schemes in ℙ². Moreover, we give a series of examples of cuspidal curves where the family V is reducible, but where π₁ (ℙ²\C) coincides (and is abelian) for all C ∈ V.