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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
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