Research Article
Maakestad H
Abstract
In previous papers the structure of the jet bundle as P-module has been studied using different techniques. In this paper we use techniques from algebraic groups, sheaf theory, generliazed Verma modules, canonical filtrations of irreducible SL(V)-modules and annihilator ideals of highest weight vectors to study the canonical filtration Ul (g)Ld of the irreducible SL(V)-module H0 (X, ï��X(d))* where X = ï��(m, m + n). We study Ul (g)Ld using results from previous papers on the subject and recover a well known classification of the structure of the jet bundle ï��l (ï��(d)) on projective space ï��(V*) as P-module. As a consequence we prove formulas on the splitting type of the jet bundle on projective space as abstract locally free sheaf. We also classify the P-module of the first order jet bundle ï��X1 (ï��X (d)) for any d ≥ 1. We study the incidence complex for the line bundle ï��(d) on the projective line and show it is a resolution of the ideal sheaf of I l (ï��(d)) - the incidence scheme of ï��(d). The aim of the study is to apply it to the study of syzygies of discriminants of linear systems on projective space and grassmannians.