Real Algebraic Varieties with Prescribed Tangent Cones

The definition of the tangent cone C(V, p) to an algebraic variety V at a point p was given by Whitney more than 40 years ago as one of the tools to get information about the geometric shape of a variety near a singular point. While the complex case has been widely and successfully studied, including from a computational point of view, only recently have some first attempts been made to elucidate the situation in the real case. For example in [O-W-3] (see also [O-W-1] and [O-W-2]), theorems are proven relating the tangent cone of a surface in R3 to its Nash fiber (the set of limits of tangent spaces at smooth points), and many examples are presented showing how the real case differs from the complex case. Since the tangent cone to a real algebraic variety is a semialgebraic set, a question which, in our opinion, is very natural is that of investigating which semialgebraic cones of Rn can be realized as tangent cones to real algebraic subsets of Rn. Partial results in this direction were proven in [F-F]; there it is shown that any closed semialgebraic cone of codimension at least one in Rn admitting a presentation with only “few” polynomial inequalities is the tangent cone to some real algebraic variety in Rn. In particular, this holds for every semialgebraic cone of codimension at least one in R3. In this paper we show that the same result is true in general for all closed semialgebraic cones of codimension at least one, without any restrictive hypothesis on the number of inequalities. Actually this is obtained as a corollary of a more general result stating that any closed semialgebraic semicone (i.e., a union of rays) of codimension at least one in Rn is the tangent semicone (i.e., a union of limits of secant rays) to a suitable real algebraic variety in Rn.