As we know, there are classes of continued fractions, finite CF / infinite CF, periodic CF/ nonperiodic CF, bounded/ unbounded. Or do classification according to the number it corresponds.

And we know solution of singularities corresponds CF.

So what is the corresponding relations between classes of CF and the singularities? For example, to periodic CF/ nonperiodic CF, what classes singularities to correspond?

  • $\begingroup$ I am not sure I understand your question, but toric geometry give explicit link between singularities of toric surfaces and continued fractions, see chapter 7 (on surfaces) in the book by Cox, Little and Schenck, "Toric varieties". $\endgroup$ – Nicolas Hemelsoet Nov 1 '17 at 9:42
  • $\begingroup$ @NicolasHemelsoet thanks, but are you it is on chapter 7? $\endgroup$ – XL _at_China Nov 1 '17 at 12:28
  • $\begingroup$ Maybe they changed the number I am not sure (I don't have the book with me) but it is in the chapter about toric surfaces. Alternatively, researching "toric surfaces continued fractions" give you lot of results as well. $\endgroup$ – Nicolas Hemelsoet Nov 1 '17 at 12:46
  • $\begingroup$ @NicolasHemelsoet thank you, I will browse the book. $\endgroup$ – XL _at_China Nov 2 '17 at 2:31

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