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, iX(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 il (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 iX1 (iX (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)

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Summary

Introduction

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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