Don O'Shea - Limits of tangent spaces to real surfaces

There is a lovely theory, due to Henry, Le, Teissier and others, describing the space of limits of tangent spaces at singular points of complex surfaces. This space encodes much delicate geometric information about the surface in a neighbourhood of the singular point. Such a theory would be highly desirable over the reals, but the complex results do not specialize in a straightforward manner. In the talk, we describe the complex theory briefly, and discuss some recent work which has cleared up a number of open questions in the real case.

In particular, let X,0 be a real surface in . We investigate the tangent semicone C⁺ to the surface (by which we mean the set of all vectors which can be obtained as a limit of a sequence t_ix_i with t_i > 0 and where the x_i tend to 0) and the Nash space K of the surface (the set of all planes which can be obtained as a limit of tangent planes to X at smooth points of xtending to 0).

We prove a structure theorem for K analogous to, but different in some interesting respects, than that over the complexes established by Lê. We show that there is a sharp (and, to us, unexpected) dichotomy between exceptional rays in C⁺ which are tangent to the singular locus of X and those that aren't. In the latter case, we determine precisely when a ray lying in the singular part of C⁺ must be exceptional, and show that the set of elements in K containing the exceptional ray cannot contain discrete elements--in fact, we can give a lower bound on the size of this set. This suggests a possible algorithm for determining when and where C⁺ (and the geometric tangent cone ) fails to be algebraic and, more speculatively, an algorithm for computing C⁺.