Slanted Vector Fields for Jet Spaces

TL;DR

This paper develops new methods to construct low pole order frames of slanted vector fields on jet spaces of complete intersections and hypersurfaces, improving pole order bounds and understanding of the loci where global generation fails.

Contribution

It introduces reformulations, a new formalism of geometric jet coordinates, and building-block vector fields to improve pole order bounds and analyze jet space structures.

Findings

Pole order bound improved to 5k-2 from k^2+2k

Constructed low pole order frames on vertical jet spaces

Identified loci where global generation fails

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 $k\geqslant1$ into a very analog of the much easier case where $k=0$, i.e. where no jet coordinates are needed.