Open AccessFoliations singular along a curve
Author(s) -
Vainsencher Israel
Publication year - 2015
Publication title -
transactions of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.43
H-Index - 7
ISSN - 2052-4986
DOI - 10.1112/tlms/tlv004
Subject(s) - foliation (geology) , mathematics , gravitational singularity , dimension (graph theory) , pure mathematics , rank (graph theory) , polynomial , space (punctuation) , vector bundle , degree (music) , mathematical analysis , combinatorics , computer science , physics , geochemistry , acoustics , metamorphic rock , geology , operating system
A general one‐dimensional foliation in the complex projective space has finitely many singularities. For an appropriately good family of subschemes inℙ n , we study the loci in the space of foliations of degree d defined by the requirement that the singularities contain a member of the family. We give a formula for the dimensions of such loci. We show that their degrees are expressed by a polynomial in d . We compute it explicitly in a few examples. Next we provide a formula for the number of isolated singular points of a foliation containing a prescribed positive‐dimensional subscheme in its singular scheme under mild assumptions. We include an appendix by Steven L. Kleiman on a theorem of Bertini suitable for sections of vector bundles with rank equal to the dimension of the base.