Geodesic
\'Eventails associ\'es \`a~des fonctions analytiques
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Monodromy in problems of algebraic geometry and differential equations, Tome 238 (2002), pp. 70-80.

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Let $X$ be a complex analytic manifold, $(f_1,\dots, f_p)$ be analytic functions on $X$, and denote by $F=f_1\dots f_p$ their product. Given a regular holonomic $\mathcal D_X$-module $\mathcal M$ and a section $m\in\mathcal M$, one can associate to the characteristic variety of the $\mathcal D_X[s_1,\ldots,s_p]$-module $\mathcal D_X[s_1,\ldots ,s_p]m f_1^{s_1}\dots f_p^{s_p}$ a finite set $\mathcal H_{f,m}$ of hyperplanes in $\mathbf C^p$. We study this characteristic variety and prove that the set $\mathcal H_{f,m}$ is contained in the union of the coordinate hyperplanes of $\mathbf C^p$ if and only if the morphism $f:\mathbf C^n \rightarrow \mathbf C^p$ has no blowing up in codimension zero and its critical locus is contained in the set $F=0$. 
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J. Briançon; Ph. Maisonobe; M. Merle. \'Eventails associ\'es \`a~des fonctions analytiques. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Monodromy in problems of algebraic geometry and differential equations, Tome 238 (2002), pp. 70-80. http://geodesic.mathdoc.fr/item/TM_2002_238_a2/

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