Bounding the regularity of subschemes invariant under Pfaff fields on projective spaces

Abstract

A Pfaff field on ${\mathbb{P}}_k^n$ is a map $\eta :{\Omega }_{{\mathbb{P}}_k^n}^s\to \mathcal{L}$ from the sheaf of differential $s$-forms to an invertible sheaf. The interesting ones are those arising from a Pfaff system, as they give rise to a distribution away from their singular locus. A subscheme $X\subseteq {\mathbb{P}}_k^n$ is said to be invariant under $\eta$ if $\eta$ induces a Pfaff field ${\Omega }_X^s\to \mathcal{L}{|}_X$. We give bounds for the Castelnuovo–Mumford regularity of invariant complete intersection subschemes (more generally, arithmetically Cohen–Macaulay subschemes) of dimension $s$, depending on how singular these schemes are, thus bounding the degrees of the hypersurfaces that cut them out.