Geodesic
On divisorial filtrations on sheaves
Matematičeskie zametki, Tome 79 (2006) no. 6, pp. 825-837.

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In this paper, we generalize the notion of Poincaré series of a multi-index divisorial filtration corresponding to a collection of sigma-processes to the case of an arbitrary locally-free sheaf on the space of blow-ups of the complex plane $\mathbb C^2$. For an arbitrary sheaf, we establish a representation of the series in terms of topological invariants of the sheaf. In particular, for the sheaf of functions, this representation coincides with the Poincaré series obtained by Gusein-Zade and Delgado. 
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     title = {On divisorial filtrations on sheaves},
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E. A. Gorskii. On divisorial filtrations on sheaves. Matematičeskie zametki, Tome 79 (2006) no. 6, pp. 825-837. http://geodesic.mathdoc.fr/item/MZM_2006_79_6_a1/

[1] Delgado F., Gusein-Zade S. M., “Poincaré series for several plane divisorial valuations”, Proc. Edinburgh Math. Soc., 46 (2003), 501–509 | DOI | Zbl

[2] Campillo A., Delgado F., Kiyek K., “Gorenstein property and symmetry for one-dimensional local Cohen-Macaulay rings”, Manuscripta Math., 83:3–4 (1994), 405–423 | DOI | MR | Zbl