2014/04/30 by Ariel Molinuevo
Mathematics · #Advanced Differential Equations and Dynamical Systems #Codimension #Commutative Algebra and Its Applications #Differential (mechanical device) #Holomorphic and Operator Theory #Hypersurface #Logarithm #Order (exchange) #Polynomial #Set (abstract data type) #Space (punctuation) #math.AG
paper · pdf · doi:10.5802/aif.3044
published as Annales de l'institut Fourier, 66 no. 4 (2016), p. 1583-1613 · Final version. 25 pages
openalex publication_date 2016/05/03 · openalex created_date 2016/06/24 · arxiv created 2016/08/13 · arxiv updated 2016/08/16 · openalex updated_date 2026/08/05
We study codimension one foliations in projective space <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>ℙ</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:math> over <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ℂ</mml:mi> </mml:math> by looking at its first order perturbations: unfoldings and deformations. We give special attention to foliations of rational and logarithmic type. For a differential form <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ω</mml:mi> </mml:math> defining a codimension one foliation, we present a graded module <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>𝕌</mml:mi> <mml:mo>(</mml:mo> <mml:mi>ω</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , related to the first order unfoldings of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ω</mml:mi> </mml:math> . If <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ω</mml:mi> </mml:math> is a generic form of rational or logarithmic type, as a first application of the construction of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>𝕌</mml:mi> <mml:mo>(</mml:mo> <mml:mi>ω</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , we classify the first order deformations that arise from first order unfoldings. Then, we count the number of isolated points in the singular set of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ω</mml:mi> </mml:math> , in terms of a Hilbert polynomial associated to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>𝕌</mml:mi> <mml:mo>(</mml:mo> <mml:mi>ω</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . We review the notion of regularity of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ω</mml:mi> </mml:math> in terms of a long complex of graded modules that we also introduce in this work. We use this complex to prove that, for generic rational and logarithmic foliations, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ω</mml:mi> </mml:math> is regular if and only if every unfolding is trivial up to isomorphism.