Unobstructedness of filling secants and the Gruson-Peskine general projection theorem

We prove an unobstructedness result for deformations of subvarieties constrained by intersections with another, fixed subvariety. We deduce smoothness and expected-dimension results for multiple-point loci of generic projections, mainly from a point or a line, or for fibres of embedding dimension 2 or less. The current version is a substantial enhancement of the previous one, including for the first time results on projection from a line

On uncountable hypersimple unidimensional theories

We extend a dichotomy between 1-basedness and supersimplicity proved in a previous paper. The generalization we get is to arbitrary language, with no restrictions on the topology (we do not demand type-definabilty of the open set in the definition of essential 1-basedness). We conclude that every (possibly uncountable) hypersimple unidimensional theory that is not s-essentially 1-based by means of the forking topology is supersimple. We also obtain a strong version of the above dichotomy in the case where the language is countable

Structure of the cycle map for Hilbert schemes of families of nodal curves

We study the relative Hilbert scheme of a family of nodal (or smooth) curves, over a base of arbitrary dimension, via its (birational) cycle map, going to the relative symmetric product. We show the cycle map is the blowing up of the discriminant locus, which consists of cycles with multiple points. We determine the relevant cotangent sheaves and complexes. We determine the structure of certain projective bundles called node scrolls, which play an important role in the geometry of Hilbert schemes.Comment: To appear in Israel J. Math. arXiv admin note: text overlap with arXiv:0803.451

On special partial types and weak canonical bases in simple theories

We define a notion of a weak canonical base for a partial type. This notion is weaker than the usual canonical base for an amalgamation base. We prove that certain family of partial types have a weak canonical base. This family clearly contains the class of amalgamation bases