On the geometry of some strata of uni-singular curves

Abstract

We study geometric properties of linear strata of uni-singular curves. The singularities of closures of the strata are resolved and the resolutions are represent as projective bundles. This enables 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 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 curves with two singular points, one of them being a node.