On the geometry of the singular locus of a codimension one foliation in ${\mathbb{P}}^n$

Abstract

We will work with codimension one holomorphic foliations over the complex projective space, represented by integrable forms $ω\in H^0(Ω^1_{\PP^n}(e))$. Our main result is that, under suitable hypotheses, the Kupka set of the singular locus of $ω\in H^0(Ω^1_{\PP^3}(e))$, defined algebraically as a scheme, turns out to be arithmetically Cohen-Macaulay. As a consequence, we prove the connectedness of the Kupka set in $\PP^n$, and the splitting of the tangent sheaf of the foliation, provided that it is locally free.

arXiv comment20 pages. One of the main theorems has been corrected

References in corpus (2)

• Stability of Holomorphic Foliations with Split Tangent Sheaf
• On the Singular Scheme of Split Foliations

Cited by in corpus (1)

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