Journal of Generalized Lie Theory and Applications

Jet Bundles on Projective Space II

H Maakestad

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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 $\mathrm{SL}(V)$-modules and annihilator ideals of highest weight vectors to study the canonical filtration $U_l(\mathfrak{g})L^d$ of the irreducible $\mathrm{SL}(V)$-module $\mathrm{H}^0 (X,\mathcal{O}_X(d))^*$ where $X = \mathbb{G}(m, m + n)$. We study $U_l(\mathfrak{g})L^d$ using results from previous papers on the subject and recover a well known classification of the structure of the jet bundle $\mathcal{P}^l(\mathcal{O}(d))$ on projective space $\mathcal{P}^l(\mathcal{O}_X(V*))$ 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 $\mathcal{P}_X^1(\mathcal{O}_X(d))$ for any $d ≥ 1$. We study the incidence complex for the line bundle $\mathcal{O}(d)$ on the projective line and show it is a resolution of the ideal sheaf of $I^l (\mathcal{O}_X(d))$ - the incidence scheme of $\mathcal{O}_X(d)$. The aim of the study is to apply it to the study of syzygies of discriminants of linear systems on projective space and grassmannians.

Article information

J. Gen. Lie Theory Appl., Volume 10, Number S2 (2016), 13 pages.

First available in Project Euclid: 16 November 2016

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Zentralblatt MATH identifier

Algebraic group Jet bundle Grassmannian P-module Generalized verma module Higher direct image Annihilator ideal Canonical filtration Discriminant Koszul complex Regular sequence Resolution


Maakestad, H. Jet Bundles on Projective Space II. J. Gen. Lie Theory Appl. 10 (2016), no. S2, 13 pages. doi:10.4172/1736-4337.1000S2-001.

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