Abstract
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 \(k\geqslant 1\) into a very analog of the much easier case where \(k=0\), i.e. where no jet coordinates are needed.
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Acknowledgments
I want to thank Jérémy Guéré for suggestions concerning presentation and Christophe Mourougane for interesting discussions on geometric jet coordinates. I warmly thank Jean-Pierre Demailly for friendly explanations in his office, which influenced the definition of the geometric jet coordinates. Lastly, I would like to gratefully thank my thesis advisor Joël Merker for his support, his very careful reading and the proposal of relevant lines of thinking.
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Darondeau, L. Slanted vector fields for jet spaces. Math. Z. 282, 547–575 (2016). https://doi.org/10.1007/s00209-015-1553-1
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DOI: https://doi.org/10.1007/s00209-015-1553-1
Keywords
- Slanted vector fields
- Geometric jet coordinates
- Logarithmic jets
- Variational method of Voisin–Siu
- Hyperbolicity
- Building-block vector fields