Collections > UNC Chapel Hill Undergraduate Honors Theses Collection > Exploring Interpolation Techniques in Enumerative Geometry

In this paper we examine the invariant subsets A of a vector space V , when we act by a group, G. Gathering some of the information about A in the equivariant cohomology ring H*G(V) is an important area of study in enumerative geometry. The descriptions we seek to find for an invariant subspace satisfies certain universal properties. A notable example of this is given a vector bundle with fiber V and structure group G over a compact manifold M looking at some cohomological data for A-points of a section can be done in the equivariant cohomology ring instead of the cohomology ring of M. The fundamental cohomology class and Segre-Schwartz- MacPherson class are two such objects that are the same in both cohomology rings. Since these classes are universal in the above sense, we expect that it is very difficult to determine the fundamental class and the Chern-Schwartz-MacPherson class. However, it is possible in certain situations. In this paper we find the fundamental class and the CSM/SSM classes in the equivariant cohomology rings for two separate groups actions. Before we make our way through calculations, we give some intuitive definition of what these classes are. The basis for the computations we do is from interpolation characterizations of the CSM class and the fundamental classes, which are stated in a way applicable to our situation. We then work through the computation one of these for two seperate group actions. The culmination of the first section is the statement and the proof of a well-known result called the Porteous identity. The second section ends with some conjectures based on the lower-dimensional results.