Commentarii Mathematici Helvetici


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Volume 86, Issue 4, 2011, pp. 947–965
DOI: 10.4171/CMH/244

Published online: 2011-09-22

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

Joana D. A. Cruz[1] and Eduardo Esteves[2]

(1) Universidade Federal de Juiz de Fora, Brazil
(2) Instituto de Matemática Pura e Aplicada, Rio de Janeiro, Brazil

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.

Keywords: Pfaff systems, projective spaces, invariant schemes, regularity

Cruz Joana, Esteves Eduardo: Bounding the regularity of subschemes invariant under Pfaff fields on projective spaces. Comment. Math. Helv. 86 (2011), 947-965. doi: 10.4171/CMH/244