On the singular scheme of split foliations

Abstract

Abstract. We prove that the tangent sheaf of a codimension one locally free distribution splits as a sum of line bundles if and only if its singular scheme is arithmetically Cohen-Macaulay. In addition, we show that a foliation by curves is given by an intersection of generically transversal holomorphic distributions of codimension one if and only if its singular scheme is arithmetically Buchsbaum. Finally, we establish that these foliations are determined by their singular schemes, and deduce that the Hilbert scheme of certain arithmetically Buchsbaum schemes of codimension 2 is birational to a Grassmannian.

On classification and moduli spaces of holomorphic distributions

“…Foliations by curves which are given by a global complete intersection of two codimension one distributions in P 3 appear in [13], where Chang provides a characterization of certain class of arithmetically Buchsbaum scheme of curves on P 3 , see [16,Corollary 3].…”

Codimension one distributions of degree 2 on the projective three-space

“…Techniques from algebraic geometry have been extremely useful in the study of singular holomorphic foliations, see for instance [1,2,5,7,9,14,15,25]. In particular, Jouanolou classified codimension one foliations on P 3 of degrees 0 and 1 in his monograph [14]; Cervau and Lins Neto showed in [3] that there exist six irreducible components of foliations of degree 2 on projective spaces; and Polishchuck, motivated by the study of holomorphic Poisson structures, also found in [20] a classification of foliations of degree 2 on P 3 under certain hypotheses of the singular set of foliations.…”