The space of triangles, vanishing theorems, and combinatorics

Abstract

We consider compactications of ( P n ) 3 n S ij , the space of triples of distinct points in projective space. One such space is a singular variety of congurations of points and lines; another is the smooth compactication 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 akey role in our analysis.