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?
$\begingroup$
$\endgroup$
1
-
$\begingroup$ 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 :-) $\endgroup$– Francesco PolizziCommented Apr 4, 2013 at 6:55
Add a comment
|
2 Answers
$\begingroup$
$\endgroup$
1
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.
answered Apr 4, 2013 at 0:49
-
2$\begingroup$ I vaguely recall this version appearing in Fulton's "Introduction to Toric Varieties". $\endgroup$ Commented Apr 4, 2013 at 0:58
$\begingroup$
$\endgroup$
1
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.
answered Apr 3, 2013 at 20:32
-
$\begingroup$ @Charles,it is only permissible to choose one post as answer,so I have chosen the one above. $\endgroup$ Commented Apr 4, 2013 at 11:08