Journal of Generalized Lie Theory and Applications

Jet Bundles on Projective Space II

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 $\mathrm{SL}(V)$-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

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

Dates
First available in Project Euclid: 16 November 2016

https://projecteuclid.org/euclid.jglta/1479265220

Digital Object Identifier
doi:10.4172/1736-4337.1000S2-001

Mathematical Reviews number (MathSciNet)
MR3663970

Zentralblatt MATH identifier
1190.58003

Citation

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. https://projecteuclid.org/euclid.jglta/1479265220