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We show that the cohomology ring of Hilbert scheme of n-points in the affine plane is isomorphic to the coordinate ring of a Gm-fixed point scheme of the n-th symmetric product of C2 for a natural Gm-action on it. This result can be seen as an analogue of a theorem of DeConcini-Procesi and Tanisaki on a description of the cohomology ring of Springer fiber of type A. We also prove similar results for type A S3-varieties and smooth hypertoric varieties. These results can be formulated in terms of the symplectic duality of Braden-Licata-Proudfoot-Webster. [ABSTRACT FROM AUTHOR]

When a fixed algebraic variety in a multiplicative group variety is intersected with the union of all algebraic subgroups of fixed dimension, a key role is played by what we call the anomalous subvarieties. These arise when the algebraic variety meets translates of subgroups in sets larger than expected. We prove a Structure Theorem for the anomalous subvarieties, and we give some applications, emphasizing in particular the case of codimension two. We also state some related conjectures about the boundedness of absolute height on such intersections as well as their finiteness. [ABSTRACT FROM PUBLISHER]