Abstract. In [Kei03], [Kei05a] and [Kei05b] we gave numerical conditions which ensure that an equisingular family is irreducible respectively T-smooth. Combining results from [GLS01] and an idea from [ChC99] we give in the present paper series of examples of families of irreducible curves on surfaces in P 3 C with only ordinary multiple points which are reducible and where at least one component does not have the expected dimension. The examples show that for families of curves with ordinary multiple points the conditions for T-smoothness in [Kei05b] have the right asymptotics. Throughout this article Σ will denote a smooth projective surface in P3 of degree C n ≥ 2, and H will be a hyperplane section of Σ. For a positive integer m we denote by Mm the topological singularity type of an ordinary m-fold point, i. e. the singularity has m smooth branches with pairwise different tangents. And for positive integers d and r we denote by V irr |dH | (rMm) the family of irreducible curves in the linear system |dH | with precisely r singular points all of which are ordinary m-fold points. V irr |dH | (rMm) is called T-smooth if it is smooth of the expected dimension expdim ( V irr |dH|(rMm) ) = dim |dH | − r · m2 + m − 4

Abstract. In [GLS01] the authors gave a general sufficient numerical condition for the T-smoothness (smoothness and expected dimension) of equisingular families of plane curves. This condition involves a new invariant γ ∗ for plane curve singularities, and it is conjectured to be asymptotically proper. In [Kei04], similar sufficient numerical conditions are obtained for the T-smoothness of equisingular families on various classes surfaces. These conditions involve a series of invariants γ ∗ α, 0 ≤ α ≤ 1, with γ ∗ 1 = γ ∗. In the present paper we compute (respectively give bounds for) these invariants for semiquasihomogeneous singularities. When studying numerical conditions for the T-smoothness of equisingular families of curves, new invariants of plane curve singularities V (f) ⊂ (�2, 0) turn up. These invariants are defined as the maximum of a function depending on the codimension of complete intersection ideals containing the Tjurina ideal, respectively the equisingularity ideal, of f, and on the intersection multiplicity of f with elements of the complete intersection ideals. In Section 1 we will define these invariants, and we will calculate them for several classes of singularities, the main results being Proposition 11, Proposition 12 and Proposition 13. It is the upper bound in Lemma 8 which ensures that the conditions for T-smoothness with these new conditions (see [GLS00], [GLS01], [Kei04]) improve than the previously known ones (see [GLS97]). In the remaining sections we introduce some notation and we gather some necessary, though mainly well-known technical results used in the proofs of Section 1. We should like to point out that the definition of the invariant γ ∗ 1 below is a modification of the invariant “γ ∗ ” defined in [GLS01], and it is always bound from above by the latter. Moreover, the latter can be replaced by it in the conditions of [GLS01]

We study geometric properties of linear strata of uni-singular curves. We resolve the singularities of closures of the strata and represent the resolutions as projective bundles. This enables us to study their geometry. In particular we calculate the Picard groups of the strata and the intersection rings of the closures of the strata. The rational equivalence classes of some geometric cycles on the strata are calculated. As an application we give an example when the proper stratum is not affine. As an auxiliary problem we discuss the collision of two singular points, restrictions on possible resulting singularity types and solve the collision problem in several cases. Then we present some cases of enumeration of

We study equi-singular strata of curves with two singular points of prescribed types. The method of the previous work [Kerner04] is generalized to this case. This allows to solve the enumerative problem for plane curves with two singular points of linear singularity types. In the general case this reduces the enumerative questions to the problem of collision of the two singular points. The method is applied to several cases, e.g. enumeration of curves with two ordinary multiple points, with a point of a linear singularity type and a node etc. Explicit numerical results are given. An elementary application of the method is the determination of Thom polynomials for curves with one singular point (for some series of singularity types). Some examples are given. MSC: primary-14N10, 14N35 secondary-14H10, 14H50