2024/08/19 by Juliana Pereira, Pereira, J. V., Jorge Santos +1
Computer Science · Mathematics · #13C05 #14F10 #14J60 #14L10 #17B45 #32M25 #Advanced Mathematical Identities #Algebraic Geometry (math.AG) #Bayesian Methods and Mixture Models #FOS: Mathematics #Statistical Distribution Estimation and Applications
paper · pdf · doi:10.48550/arxiv.2408.10057
openalex publication_date 2024/08/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In the paper Stability of Holomorphic Foliations with Split Tangent Sheaf one finds a study of the locus Dec where the tangent sheaf of a \it family of foliations in ℙ\mathbb Cn is \it decomposable, i.e. a sum of line bundles. A prime conclusion is an ``openness'' result: once the singular locus has sufficiently large codimension, Dec turns out to be open. In the present paper, we study the locus LF of points of a family of distributions where the tangent sheaf is \it locally free. Through general Commutative Algebra, we show that LF is open provided that singularities have codimension at least three. When dealing with foliations rather than distributions, the condition on the lower bound of the singular set can be weakened by the introduction of ``Kupka'' points. We apply the available ``openness'' results to families in ℙ\mathbb Cn and in \mathcal B, the variety of Borel subgroups of a simple group. By establishing a theorem putting in bijection irreducible components of the space of two-dimensional subalgebras of a given semi-simple Lie algebra and its nilpotent orbits, we conclude that the space of foliations of \it rank two on ℙ\mathbb Cn and \mathcal B, may have quite many irreducible components as n and dim \mathcal B grow. We also set in place several algebro-geometric foundations for the theory of families of distributions in two appendices.