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Higher dimensional analogues of periodic continued fractions and cusp singularities. (English) Zbl 0585.14004
There is a well-known relationship between periodic continued fractions and 2-dimensional cusp singularities. Let $$\pi$$ : $$U\to V$$ be the minimal resolution of a 2-dimensional cusp singularity (V,p). Then the exceptional set $$X=\pi^{-1}(p)$$ is either a cycle of s rational curves with self-intersection numbers $$a_ 1,a_ 2,...,a_ s\leq -2$$ at least one of which is strictly smaller than -2 (s$$\geq 2)$$, or a rational curve with a node and with a self-intersection number $$a<0$$. Then we can associate to it the periodic continued fraction
$$\omega =[[\overline{-a_ 1,-a_ 2,...,-a_ s}]]=(-a_ 1)-\underline 1| \overline{(-a_ 2)}-...-\underline 1/\overline{(-a_ s)}- \underline 1| \overline{(-a_ 1)}-...,$$
or
$$\omega =[[\overline{-a+2}]]=(-a+2)-\underline 1| \overline{(-a+2)}- \underline 1| \overline{(-a+2)}....$$
Conversely, we can construct a 2-dimensional cusp singularity and its resolution as above, from a periodic continued fraction $$\omega$$ first by constructing a convex cone in $${\mathbb{R}}^ 2$$ and then applying the theory of torus embeddings. Moreover, the dual graph of X can be thought of as a subdivision of a circle $$S^ 1$$, with $$a_ 1,a_ 2,...,a_ n$$ attached to s vertices as weights in this order. In this paper, we generalize the above relationship to higher dimensions and construct higher dimensional cusp singularities from suitable analogues of periodic continued fractions. The well-known Hilbert modular cusp singularities are special cases of the cusp singularities we obtain.

##### MSC:
 14B05 Singularities in algebraic geometry 11A55 Continued fractions 14H20 Singularities of curves, local rings
##### Keywords:
cusp singularities; periodic continued fraction
Full Text:
##### References:
 [1] A. ASH, D. MUMPORD, M. RAPOPORT AND Y. TAI, Smooth compactification of locally symmetric varieties, Math. Sci. Press, Brookline, Mass., 1975. · Zbl 0334.14007 [2] C. BANICA AND O. STANASJLA, Algebraic methods in the grobal theory of complex spaces, Editura Academiei, Bucuresti and John Wiley & Sons, London, New York, Sydney and Toronto, 1976. [3] H. COHN, Finiteness of the formal ring of a quadratic sector, J. of Number Theory (1976), 206-217. · Zbl 0326.12002 · doi:10.1016/0022-314X(76)90102-5 [4] H. COHN, Formal ring of a cubic solid angle, J. of Number Theory 10 (1978), 35-50 · Zbl 0382.10022 · doi:10.1016/0022-314X(78)90032-X [5] M. DEMAZURE, Sous-groupes algebriques de rang maximum du groupe de Cremona, Ann Sci. Ecole Norm. Sup., Paris, (4) 3 (1970), 507-588. · Zbl 0223.14009 · numdam:ASENS_1970_4_3_4_507_0 · eudml:81870 [6] E. FREITAG, Lokale und globale Invarianten der Hilbertschen Modulgruppe, Invent. Math 17 (1972), 106-143. · Zbl 0272.32010 · doi:10.1007/BF01418935 · eudml:142163 [7] A. GROTHENDIECK, Sur quelques points d’algebre homologique, Thoku Math. J. 9 (1957), 119-227. · Zbl 0118.26104 [8] A. GROTHENDIECK AND J. A. DIEUDONNE, Elements de Geometrie Algebrique I, Springer Verlag, Berlin, Heidelberg, New York, 1971. [9] R. C. GUNNING AND H. Rossi, Analytic functions of several complex variables, Prentice Hall, Englewood Cliff, N. J., 1965. · Zbl 0141.08601 [10] F. HIRZEBRUCH, Hubert modular surfaces, Lns. Math. 19 (1973), 183-281 · Zbl 0285.14007 [11] F. HIRZEBRUCH AND VAN DERGEER, Lectures on Hubert modular surfaces, Les Presse de L’Universite de Montreal, 1981. [12] M. -N. ISHIDA, Torus embeddings and dualizing complexes, Thoku Math. J. 32 (1980), 111-146. · Zbl 0454.14021 · doi:10.2748/tmj/1178229687 [13] M. -N. ISHIDA, Tsuchihashi’s cusp singularities are Buchsbaum singularities, · Zbl 0572.14004 [14] G. KEMPF, F. KNUDSEN, D. MUMFORD AND B. SAINT-UONAT, Toroidal embeddings I, Lecture Notes in Math. 339, Springer-Verlag, Berlin, Heidelberg, New York, 1972. [15] I. NAKAMURA, Inoue-Hirzebruch surfaces and a duality of hyperbolic unimodular singula rities I, Math. Ann. 252 (1980), 221-235. · Zbl 0425.14010 · doi:10.1007/BF01420085 · eudml:163458 [16] T. ODA, Lectures on torus embeddings and applications (Based on joint work with K. Miyake), Tata. Inst. of Fund. Res., Bombay, 1978. · Zbl 0417.14043 [17] R. T. ROCKAFELLAR, Convex analysis, Princeton University Press, 1965 · Zbl 0932.90001 [18] P. SCHENZEL, Applications of dualizing complexes to Buchsbaum rings, Advances in Math 44 (1982), 61-77. · Zbl 0492.13011 · doi:10.1016/0001-8708(82)90065-2 [19] E. THOMAS AND A. T. VASQUEZ, On the resolution of cusp singularities and the Shintan decomposition in totally real cubic number fields, Math. Ann. 247 (1980), 1-20. · Zbl 0403.14005 · doi:10.1007/BF01359864 · eudml:163345 [20] H. TSUCHIHASHI, 2-dimensional periodic continued functions and 3-dimentional cus singularities, Proc. Japan Acad., 58(A) (1982), 262-264. · Zbl 0534.14006 · doi:10.3792/pjaa.58.262 [21] E. B. VINBERG, Theory of homogeneous convex cones, Trans. Moscow Math. Soc. 1 (1967), 303-368. · Zbl 0138.43301 [22] K. WATANABE, On pluri genera of normal isolated singularities I, Math. Ann. 250 (1980), 65-94. · Zbl 0414.32005 · doi:10.1007/BF02599787 · eudml:163411 [23] J. L. BRYLINSKI, Decomposition simpliciale d’un reseau invariante par un groupe fin d’automorphismes, C. R. Acad. Sci. Paris 288 (1979), 137-139. · Zbl 0406.14022 [24] F. EHLERS, Eine Klasse komplexer Mannigfaltigkeiten und die Auflosung einiger isolierte Singularitaten, Math. Ann. 218 (1975), 127-156. · Zbl 0301.14003 · doi:10.1007/BF01370816 · eudml:162789 [25] I. SATAKE, Appendix to ”On numerical invariants of the arithmetic quotient spaces o Q-rank one, ”
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