Analysis Seminar

Speaker: Pinaki Mondal (Toronto)

"On projective completions of C^n"

Time: 13:00

Room: MC 108

A projective completion of Cn is an open immersion phi:Cn -> X, where X is a projective variety. This work is directed towards a study of completions which are "good" for a given polynomial map P:Cn -> Cn with all fibers finite, in the sense that for an open dense set of points (a_1,...,an) in Cn the intersection of the closure in the completion of hypersurfaces {P_j(z)= a_j} in Cn, corresponding to n coordinate functions P_j of the map P, has only points in the original Cn (i.e, there are no points at infinity in the intersection of the closures of those hypersurfaces). We start with the observation that "filtrations" on the polynomial ring of n variables give rise to a class of projective completions of Cn. We prove that there are completions of this type which are good in the above sense. Then we consider special types of filtrations which are given by a finite number of "generalized degrees". It turns out that there are good completions which arise from this type of filtrations. The completions of this type are nicer in the sense that we can say a bit more about the structure of the parts of the variety at infinity. In this talk I will go over the idea of the proofs, giving some examples along the way.