Open Access
2016 Jet Bundles on Projective Space II
H Maakestad
J. Gen. Lie Theory Appl. 10(S2): 1-13 (2016). DOI: 10.4172/1736-4337.1000S2-001


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.


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H Maakestad. "Jet Bundles on Projective Space II." J. Gen. Lie Theory Appl. 10 (S2) 1 - 13, 2016.


Published: 2016
First available in Project Euclid: 16 November 2016

zbMATH: 1190.58003
MathSciNet: MR3663970
Digital Object Identifier: 10.4172/1736-4337.1000S2-001

Keywords: algebraic group , annihilator ideal , Canonical filtration , discriminant , Generalized verma module , Grassmannian , Higher direct image , jet bundle , Koszul complex , P-module , regular sequence , resolution

Rights: Copyright © 2016 Ashdin Publishing (2009-2013) / OMICS International (2014-2016)

Vol.10 • No. S2 • 2016
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