Codimension one holomorphic distributions on the projective three-space

Abstract

We study codimension one holomorphic distributions on the projective three-space, analyzing the properties of their singular schemes and tangent sheaves. In particular, we provide a classification of codimension one distributions of degree at most 2 with locally free tangent sheaves, and show that codimension one distributions of arbitrary degree with only isolated singularities have stable tangent sheaves. Furthermore, we describe the moduli space of distributions in terms of Grothendieck's Quot-scheme for the tangent bundle. In certain cases, we show that the moduli space of codimension one distributions on the projective space is an irreducible, nonsingular quasi-projective variety. Finally, we prove that every rational foliation, and certain logarithmic foliations have stable tangent sheaves.

43 pages, to appear in International Mathematics Research Notices. Major changes following suggestions of the referees

References in corpus (9)

• On G-Fano threefolds
• On the Singular Scheme of Split Foliations
• Rank two globally generated vector bundles with c_1 \leq 5
• On degeneracy schemes of maps of vector bundles and applications to holomorphic foliations
• Families of distributions and Pfaff systems under duality
• On the singular scheme of codimension one holomorphic foliations in P^3
• Residues for flags of holomorphic foliations
• Absolutely $k$-convex domains and holomorphic foliations on homogeneous manifolds
• On non-Kupka points of codimension one foliations on ${\mathbb{P}}^3$

Cited by in corpus (6)

• Codimension one Fano distributions on Fano manifolds
• On holomorphic distributions on Fano threefolds
• Codimension one distributions of degree 2 on the three-dimensional projective space
• Moduli of Distributions via Singular Schemes
• Non-Ulrich representation type
• Analytic varieties invariant by foliations and Pfaff systems