The Grassmann variety G(k, h) parametrizes (k −1)-dimensional projective subspaces in P n−1. Projective subspaces are just algebraic subvarieties of degree 1. It is natural to look for parameter spaces parametrizing subvarieties of a given degree d ≥ 1. Here, however, we encounter some new phenomena. Namely, an irreducible variety can degenerate into a reducible one (e.g., a curve can degenerate into a collection of straight lines). Moreover, consider a reducible variety, say, a union of two distinct lines. Such a variety can degenerate into one line, which apparently has a smaller degree. Of course, in this case it is natural to count the limiting line with multiplicity 2. To take into account all of these possibilities, we need the notion of an algebraic cycle.
KeywordsSymmetric Polynomial Smooth Point Symmetric Product Projective Subspace Algebraic Cycle
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