Vanishing theorems and character formulas for the Hilbert scheme of points in the plane

Details Vanishing theorems and character formulas for the Hilbert scheme of points in the plane Vanishing theorems and character formulas for the Hilbert scheme of points in the plane

2002-01-16

arxiv.org/abs/math/0201148v1

Invent. Math. 149, no. 2 (2002) 371-407

Mathematics

Algebraic Geometry

33 pages

Scientific paper

10.1007/s002220200219

Earlier we showed that the Hilbert scheme of $n$ points in the plane can be identified with the Hilbert scheme of regular $S_n$ orbits on $C^{2n}$. Using this result, together with a recent theorem of Bridgeland, King and Reid on the generalized McKay correspondence, we prove vanishing theorems for tensor powers of tautological bundles on the Hilbert scheme. We apply the vanishing theorems to establish (among other things) the character formula for diagonal harmonics conjectured by Garsia and the author. In particular we prove that the dimension of the space of diagonal harmonics is equal to $(n+1)^{n-1}$.

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