2013/09/15 by Brian Pike, Pike, Brian
Mathematics · #17B66 (Primary) 32B15 #32S05 #32S25 (Secondary) #Algebraic Geometry (math.AG) #Complex Variables (math.CV) #FOS: Mathematics #math.AG #math.CV #msc:17B66 #msc:32B15 #msc:32S05 #msc:32S25
paper · pdf · doi:10.48550/arxiv.1309.3769
22 pages. From v1, improve prose, shorten a few proofs, and update contact information
arxiv created 2014/10/17 · arxiv updated 2014/10/21
The germ of an analytic set (X,p) in ℂn has an associated \mathscrOℂn,p-module Der(-log X) of `logarithmic vector fields', the ambient germs of holomorphic vector fields tangent to the smooth locus of X. For a module L⊆ Der(-log X) let Ik(L) be the ideal generated by the k× k minors of a matrix of generators for L; these are the Fitting ideals of Derℂn,p/L. We aim to: (i) find sufficient conditions on \Ik(L)\ to prove L=Der(-log X); (ii) identify \Ik(Der(-log X))\, to provide a necessary condition for equality; and (iii) provide a geometric interpretation of these ideals. Even for (X,p) smooth, an example shows that Fitting ideals alone are insufficient to prove equality, although we give a different criterion. Using (ii) and (iii) in the smooth case, we give partial answers to (ii) and (iii) for arbitrary (X,p). When (X,p) is a hypersurface, we give sufficient algebraic or geometric conditions for the reflexive hull of L to equal Der(-log X); for L reflexive, this answers (i) and generalizes criteria of Saito for free divisors and Brion for linear free divisors.