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Cusp singularities given by reflections of stellable cones. (English) Zbl 0756.14001
The author builds “Tsuchihashi cusps” [H. Tsuchihashi, Tôhoku Math. J., II. Ser. 35, 607-639 (1983; Zbl 0585.14004)] (this is a generalization of Hilbert modular cusp singularities). Such a singularity is defined by a pair $$(C,\Gamma)$$ of an open convex cone $$C\subset\mathbb{R}^ n$$ and a discrete group $$\Gamma\subset GL(n,\mathbb{Z})$$ with good conditions. The author defines and studies the notion of “semi-integral stellable polyhedral cones” $$C$$, the group $$\Gamma$$ generated by the reflections with respect to the facets of such a $$C$$ gives rise to a good pair $$(C,\Gamma)$$. There is a duality among stellable cones, the corresponding singularities are dual in the sense of Tsuchihashi [loc. cit.].
At the end, the author gives effective examples of his singularities and computes the arithmetic genus default $$\chi_ \infty$$ and the Ogata zeta zero $$Z(0)$$ and verifies on his examples the Ogata-Satake conjecture: the $$\chi_ \infty$$ of a cusp is equal to the $$Z(0)$$ of its dual. A proof of this conjecture is announced as forthcoming.
[See also: E. B. Vinberg, Math. USSR, Izv. 5(1971), 1083-1119 (1972); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 35, 1072-1112 (1971; Zbl 0247.20054) and T. Satake and S. Ogata, in Automorphic forms and geometry of arithmetic varieties, Adv. Stud. Pure Math. 15, 1-27 (1989; Zbl 0712.14009)].

##### MSC:
 14B05 Singularities in algebraic geometry 14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) 14E15 Global theory and resolution of singularities (algebro-geometric aspects)
##### Keywords:
Tsuchihashi cusps; arithmetic genus default; zeta zero
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