Principal part bundles on $\PP^n$ and quiver representations

Abstract

We study the principal parts bundles $P^k (L)$ of the degree $d$ line bundle $L$ on the $n$ dimensional projective space as homogeneous bundles and we describe their associated quiver representations. We use this approach to show that if $n$ is greater or equal that 2, and $0\leq d<k$, then there exists an invariant splitting $P^k(L)=Q\oplus (S^dV\otimes \OO_{\PP^n})$ with $Q$ a stable homogeneous vector bundle. The splitting properties of such bundles were previously known only for n=1 or $k\leq d$ or $d<0$. Moreover we show that for any $d$ and any $h<k$ the canonical map from $P^k(L)$ to $P^h(L)$ always induces a linear map on the spaces of global sections which has maximal rank.