Irreducibility of Equisingular Families of Curves

Abstract

In 1985 Joe Harris proved the long standing claim of Severi that equisingular families of nodal plane curves are irreducible 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 S it thus makes sense to look for conditions which ensure that the family V_D(S_1,...,S_r) of irreducible curves in the linear system |D| with precisely r singular points of types S_1,...,S_r is irreducible. Considering different surfaces including general surfaces in P^3 and products of curves, we produce a sufficient condition of the type deg(X(S_1))^2 +...+ deg(X(S_r))^2 < a*(D-K_S)^2, where a is some constant and X(S_i) some zero-dimensional scheme associated to the singularity type. Our results carry the same asymptotics as the best known results in this direction in the plane case, even though the coefficient is worse. For most of the considered surfaces these are the only known results in that direction.