[Paper Review] Regularity of subschemes invariant under Pfaff fields on projective spaces

Joana Darc Antonia Santos Da Cruz, Eduardo Esteves|ArXiv.org|Feb 13, 2009

Commutative Algebra and Its Applications11 references3 citations

TL;DR

This paper establishes bounds on the Castelnuovo–Mumford regularity of arithmetically Cohen–Macaulay subschemes invariant under Pfaff fields on projective spaces, showing that the regularity is controlled by the degree of the Pfaff field and the singular locus of the subscheme. For reduced, a.C.M. curves of dimension 1 invariant under a rank-1 Pfaff field with finite singular locus, the regularity is exactly $ r = m + 1 + \rho $, where $ \rho = \mathrm{reg}(\Sigma_C) - r + 2 $, generalizing earlier bounds for algebraic integrals of vector fields.

ABSTRACT

A Pfaff field on a projective space is a map from the sheaf of differential s-forms, for a certain s, 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 of the projective space is said to be invariant under the Pfaff field, if the latter induces a Pfaff field on the subscheme. 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.

Motivation & Objective

• To bound the Castelnuovo–Mumford regularity of complete intersection and arithmetically Cohen–Macaulay subschemes invariant under Pfaff fields on $ \mathbb{P}^n_k $.
• To generalize prior results on the Poincaré problem by relating regularity to the degree of the Pfaff field and the singular locus of the subscheme.
• To provide a cohomological criterion for regularity bounds using vanishing of sheaf cohomology groups on the singular locus.
• To extend known bounds for curves and hypersurfaces to higher-dimensional, singular, and non-reduced invariant subschemes.

Proposed method

• Uses the theory of Pfaff fields as maps $ \eta: \Omega^s_{\mathbb{P}^n_k} \to \mathcal{L} $, where $ \mathcal{L} $ is an invertible sheaf, with degree $ m = \deg(\mathcal{L}) + s $.
• Applies cohomological techniques, particularly the vanishing of $ H^1 $ and $ H^2 $ groups of ideal sheaves twisted by specific degrees.
• Employs exact sequences involving ideal sheaves of $ \mathcal{S} $, $ C $, and their intersections to relate cohomology to regularity.
• Relies on the regularity of the singular locus $ \mathcal{S} $ and the singular locus of the subscheme $ \Sigma_C $, using known results on regularity of zero-dimensional schemes.
• Uses the isomorphism $ \mathcal{I}_{\mathcal{S} \cap C, C}(m-1) \cong \widetilde{\Omega}^1_C $, where $ \widetilde{\Omega}^1_C $ is the torsion-free part of the canonical sheaf.
• Applies the subcanonical condition $ \omega_C \cong \mathcal{O}_C(r-3) $ and the Gorenstein property to relate $ \mathcal{I}_{\Sigma_C, C} \otimes \omega_C \cong \widetilde{\Omega}^1_C $.

Experimental results

Research questions

• RQ1 What is the maximal possible regularity of a reduced, arithmetically Cohen–Macaulay curve invariant under a rank-1 Pfaff field of degree $ m $ with finite singular locus?
• RQ2 How does the singular locus of the curve influence the regularity bound beyond the degree $ m $ of the Pfaff field?
• RQ3 Can the regularity of invariant subschemes be bounded in terms of the regularity of their singular loci and the Pfaff field’s degree?
• RQ4 Under what cohomological conditions does the regularity of an invariant subscheme equal $ m + 1 + \rho $, where $ \rho $ depends on the singular locus?
• RQ5 To what extent do the bounds for curves extend to higher-dimensional invariant subschemes?

Key findings

• For a reduced, arithmetically Cohen–Macaulay curve $ C \subseteq \mathbb{P}^n_k $ of dimension 1 invariant under a rank-1 Pfaff field $ \eta $ of degree $ m \geq 1 $, with finite singular locus, the regularity satisfies $ r = m + 1 + \rho $, where $ \rho = \mathrm{reg}(\Sigma_C) - r + 2 $.
• The bound $ r = m + 1 + \rho $ is sharp and is achieved only when the singular locus $ \Sigma_C $ is nonempty and $ r \geq m + 4 $, ensuring $ \rho \geq 3 $.
• The regularity of the singular locus $ \mathcal{S} $ of the Pfaff field is $ \mathrm{reg}(\mathcal{S}) = mn - n + 2 $, which is used to control cohomological vanishing in the proof.
• The vanishing of $ H^1(\mathcal{I}_{\mathcal{S}}(j)) $ for $ j = m + \rho - 2 $ is essential, and this holds due to the regularity bound on $ \mathcal{S} $.
• The non-vanishing of $ H^1(\mathcal{I}_{\Sigma_C, C}(\sigma - 2)) $ is shown via the non-vanishing of $ H^1(\mathcal{I}_{\Sigma_C}(\sigma - 2)) $, which follows from $ \Sigma_C $ being finite and nonempty.
• The result generalizes earlier bounds by Cerveau and Lins Neto, and by Brunella and Mendes, by extending them to singular curves and higher-dimensional settings using cohomological tools.