JournalscmhVol. 86, No. 4pp. 947–965

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

Abstract

A Pfaff field on Pkn\mathbb{P}^n_k is a map η ⁣:ΩPknsL\eta \colon \Omega^s_{\mathbb{P}^n_k} \to \mathcal{L} from the sheaf of differential ss-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 XPknX \subseteq \mathbb{P}^n_k is said to be invariant under η\eta if η\eta induces a Pfaff field ΩXsLX\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 ss, 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