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I have an impression that there is linkage or relation between singulariry of algebraic variety and continued fraction when I read some book on resolution of singularity or algebraic geometry.Could any one give some reference for that?

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You can have a look at Section 2 of my paper with E. Mistretta: arxiv.org/abs/0805.1424 where the case of dimension $2$ cyclic quotient singularities is considered. Sorry for being self-referential :-) –  Francesco Polizzi Apr 4 '13 at 6:55
@Francesco,a lot of thanks to you –  XL _at_China Apr 4 '13 at 7:17

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Continued fractions appear naturally in the resolution of quotient singularities of surfaces (and presumably in higher dimensions as well). From a topological point of view, a neighborhood of the singularity is the cone on a lens space L(p,q), and a particular continued fraction for q/p gives an explicit piece of a smooth complex surface with the same boundary. This is explained very nicely in the notes, "Differentiable Manifolds and Quadratic Forms" by Hirzebruch, Neumann, and Koh, and presumably in many algebraic geometry texts.

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I vaguely recall this version appearing in Fulton's "Introduction to Toric Varieties". –  Allen Knutson Apr 4 '13 at 0:58
@Danny,Thank you very much –  XL _at_China Apr 4 '13 at 1:27

In the case of Hilbert modular surfaces, continued fractions appeared in the work of Hirzebruch on their singularities. This is easy to find online. This generalises somewhat in Shintani's work, as was written up in Sankaran's thesis.

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@Charles,thanks a lot –  XL _at_China Apr 3 '13 at 20:38
@Charles,it is only permissible to choose one post as answer,so I have chosen the one above. –  XL _at_China Apr 4 '13 at 11:08

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