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.