The Space of Triangles, Vanishing Theorems, and Combinatorics

Details The Space of Triangles, Vanishing Theorems, and Combinatorics The Space of Triangles, Vanishing Theorems, and Combinatorics

1996-01-23

arxiv.org/abs/alg-geom/9601022v1

Journal of Algebra, Vol. 222 (1999) 17-50

Mathematics

Algebraic Geometry

LaTeX Version 2.09 <7 Dec 1989>, 38 pages, author-supplied PostScript file available at http://www.math.ruu.nl/people/vdkallen

Scientific paper

We consider compactifications of the space of triples of distinct points in projective $n$-space. One such space is a singular variety of configurations of points and lines; another is the smooth compactification of Fulton and MacPherson; and a third is the triangle space of Schubert and Semple. We compute the sections of line bundles on these spaces, and show that they are equal as GL(n) representations to the generalized Schur modules associated to ``bad'' generalized Young diagrams with three rows (Borel-Weil theorem). On the one hand, this yields Weyl-type character and dimension formulas for the Schur modules; on the other, a combinatorial picture of the space of sections. Cohomology vanishing theorems play a key role in our analysis.

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