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 Pn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {P} ^n$$\end{document} 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 Pn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {P} ^n$$\end{document} with several irreducible smooth components. Both the pole order (here =5k-2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$=5k-2$$\end{document}) and the determination of the locus where the global generation statement fails are improved compared to the literature (previously =k2+2k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$=k^{2}+2k$$\end{document}), 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⩾1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k\geqslant 1$$\end{document} into a very analog of the much easier case where k=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k=0$$\end{document}, i.e. where no jet coordinates are needed.