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

### Joana D. A. Cruz

Universidade Federal de Juiz de Fora, Brazil### Eduardo Esteves

Instituto de Matemática Pura e Aplicada, Rio de Janeiro, Brazil

## Abstract

A Pfaff field on $\mathbb{P}^n_k$ is a map $\eta \colon \Omega^s_{\mathbb{P}^n_k} \to \mathcal{L}$ 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 $X \subseteq \mathbb{P}^n_k$ is said to be invariant under $\eta$ if $\eta$ induces a Pfaff field $\Omega^s_X\to\mathcal{L} |_X$. 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.

## Cite this article

Joana D. A. Cruz, Eduardo Esteves, Bounding the regularity of subschemes invariant under Pfaff fields on projective spaces. Comment. Math. Helv. 86 (2011), no. 4, pp. 947–965

DOI 10.4171/CMH/244