Smoothness of Equisingular Families of Curves

Abstract

Francesco Severi showed that equisingular families of plane nodal curves are T-smooth, i.e. smooth of the expected dimension, whenever they are non-empty. For families with more complicated singularities this is no longer true. Given a divisor D on a smooth projective surface it thus makes sense to look for conditions which ensure that the family V of irreducible curves in the linear system |D| with precisely r singular points of types S1,...,Sr is T-smooth. Considering different surfaces including the projective plane, general surfaces in projective 3-space, products of curves and geometrically ruled surfaces, we produce a sufficient condition of the type gamma_a(S1)+...+gamma_a(Sr) < c*(D-K)^2, where gamma_a is some invariant of the singularity type and c is some constant. This generalises the results obtained by Greuel, Lossen and Shustin in math.AG/9903179 for the plane case, combining their methods and the method of Bogomolov instability, used by Chiantini and Sernesi in alg-geom/9602012. For many singularity types the $gamma_a$-invariant leads to essentially better conditions than the invariants previously used, and for most classes of geometrically ruled surfaces our results are the first known for T-smoothness at all.