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, iX(d))* where X = i(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 il (i(d)) on projective space i(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 iX1 (iX (d)) for any d ≥ 1. We study the incidence complex for the line bundle i(d) on the projective line and show it is a resolution of the ideal sheaf of I l (i(d)) - the incidence scheme of i(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.
Highlights
In a series of papers of Maakestad [1,2,3,4], the structure of the jet bundle as P-module has been studied using different techniques
Using results obtained in studies of Maakestad [1] we classify Ul (g)Ld and as a corollary we recover a well known result on the structure of the jet bundle l ( (d)) on (V*) as P-module
We classify the P-module of the first order jet bundle X1( X (d)) on any grassmannian X = (m, m + n) (Corollary 3.10)
Summary
In a series of papers of Maakestad [1,2,3,4], the structure of the jet bundle as P-module has been studied using different techniques. Using Koszul complexes and general properties of jet bundles we prove it is a locally free resolution of the ideal sheaf of I l( (d)) - the incidence scheme of (d). There is by studies of Maakestad [11] for all 1 ≤ l < d an exact sequence of locally free X-modules We study the jet bundle of any finite rank G-linearized locally free sheaf on the grassmannian G/P= (m, m + n) as Pl -module, where Pl ⊆ P is a maximal linearly reductive subgroup.
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