Principal parts bundles on projective spaces and quiver representations

Abstract

We study the principal parts bundles $\mathcal {P}^{k}\mathcal {O}_{\mathbb {P}^{n}}(d)$ as homogeneous bundles and we describe their associated quiver representations. With this technique we show that if n≥2 and 0≤d<k then there exists an invariant decomposition $\mathcal {P}^{k}\mathcal {O}_{\mathbb {P}^{n}}(d)=Q_{k,d}\oplus(S^{d}V\otimes \mathcal {O}_{\mathbb {P}^{n}})$ with Q _k,d a stable homogeneous vector bundle. The decomposition properties of such bundles were previously known only for n=1 or k≤d or d<0. Moreover we show that the Taylor truncation maps $H^{0}\mathcal {P}^{k}\mathcal {O}_{\mathbb {P}^{n}}(d)\to H^{0}\mathcal {P}^{h}\mathcal {O}_{\mathbb {P}^{n}}(d)$, defined for any h≤k and any d, have maximal rank.