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Research Article Open Access
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, Ã¯ÂÂX(d))* where X = Ã¯ÂÂ(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 Ã¯ÂÂl (Ã¯ÂÂ(d)) on projective space Ã¯ÂÂ(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 Ã¯ÂÂX1 (Ã¯ÂÂX (d)) for any d ≥ 1. We study the incidence complex for the line bundle Ã¯ÂÂ(d) on the projective line and show it is a resolution of the ideal sheaf of I l (Ã¯ÂÂ(d)) - the incidence scheme of Ã¯ÂÂ(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.
Algebraic group, Jet bundle, Grassmannian, P-module, Generalized verma module, Higher direct image, Annihilator ideal, Canonical filtration, Discriminant, Koszul complex, Regular sequence, Resolution, Lie Theory, Lie bracket, Lie algebra, Topology, Lie triple systems