Slanted Vector Fields for Jet Spaces

Low pole order frames of slanted vector fields are constructed on the space of vertical k-jets of the universal family of complete intersections in $\mathbb{P}^n$ and, adapting the arguments, low pole order frames of slanted vector fields are also constructed on the space of vertical logarithmic k-jets along the universal family of projective hypersurfaces in $\mathbb{P}^n$ with several irreducible smooth components. Both the pole order (here $=5k-2$) and the determination of the locus where the global generation statement fails are improved compared to the literature (previously $=k^2+2k$), thanks to three new ingredients; we reformulate the problem in terms of some adjoint action, we introduce a new formalism of geometric jet coordinates, and then we construct what we call building-block vector fields, making the problem for arbitrary jet order $k\geqslant1$ into a very analog of the much easier case where $k=0$, i.e. where no jet coordinates are needed.