Journal of Commutative Algebra

Monomial valuations, cusp singularities, and continued fractions

David J. Bruce, Molly Logue, and Robert Walker

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


This paper explores the relationship between real valued monomial valuations on $k(x,y)$, the resolution of cusp singularities and continued fractions. It is shown that, up to equivalence, there is a one-to-one correspondence between real valued monomial valuations on $k(x,y)$ and continued fraction expansions of real numbers between zero and one. This relationship with continued fractions is then used to provide a characterization of the valuation rings for real valued monomial valuations on $k(x,y)$. In the case when the monomial valuation is equivalent to an integral monomial valuation, we exhibit explicit generators of the valuation rings. Finally, we demonstrate that, if $\nu$ is a monomial valuation such that $\nu(x)=a$ and $\nu(y)=b$, where $a$ and $b$ are relatively prime positive integers larger than one, then $\nu$ governs a resolution of the singularities of the plane curve $x^{b}=y^{a}$ in a way we make explicit. Further, we provide an exact bound on the number of blow ups needed to resolve singularities in terms of the continued fraction of $a/b$.

Article information

J. Commut. Algebra, Volume 7, Number 4 (2015), 495-522.

First available in Project Euclid: 19 January 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 13F30: Valuation rings [See also 13A18] 14E15: Global theory and resolution of singularities [See also 14B05, 32S20, 32S45] 16W60: Valuations, completions, formal power series and related constructions [See also 13Jxx]

Monomial valuations resolution of singularities continued fractions


Bruce, David J.; Logue, Molly; Walker, Robert. Monomial valuations, cusp singularities, and continued fractions. J. Commut. Algebra 7 (2015), no. 4, 495--522. doi:10.1216/JCA-2015-7-4-495.

Export citation


  • Shreeram Abhyankar, Local uniformization on algebraic surfaces over ground fields of characteristic, Ann. Math. 63 (1956), 491–526.
  • Shreeram Shankar Abhyankar, Resolution of singularities of embedded algebraic surfaces, Academic Press, New York, 1966.
  • Daren C. Collins, Continued fractions, MIT Undergrad. J. Math. 1 (1999).
  • Vincent Cossart and Olivier Piltant, Resolution of singularities of threefolds in positive characteristic I, J. Alg. 320 (2008), 1051–1082.
  • ––––, Resolution of singularities of threefolds in positive characteristic II, J. Alg. 321 (2009), 1836–1976.
  • David A. Cox, John B. Little and Henry K. Schenck, Toric varieties, American Mathematical Society, Providence, R.I., 2011.
  • Steven Dale Cutkosky, Resolution of singularities, American Mathematical Society, Providence, R.I, 2004.
  • ––––, Resolution of singularities for $3$-folds in positive characteristic, American J. Math. 131 (2009), 59–127.
  • Charles Favre and Mattias Jonsson, The valuative tree, Springer, New York, 2004.
  • Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York, 1977.
  • Heisuke Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero: I, Ann. Math. 79 (1964), 109–203.
  • ––––, Resolution of singularities of an algebraic variety over a field of characteristic zero: II, Ann. Math. 79 (1964), 205–236.
  • Hideyuki Matsumura, Commutative ring theory, Cambridge University Press, Cambridge, 1989.
  • Janos Kollar, Lectures on resolution of singularities, Princeton University Press, Princeton, 2007.
  • Patrick Popescu-Pampu, The geometry of continued fractions and the topology of surface singularities, ArXiv Mathematics e-prints (2005), page 6432.
  • Mark Spivakovsky, Valuations in function fields of surfaces, Amer. J. Math. 112 (1990), 107–156.
  • James K Strayer, Elementary number theory, Waveland Press, Prospect Heights, IL, 2002.
  • Oscar Zariski, Local uniformization on algebraic varieties, Ann. Math. 41 (1940), 852–896.
  • ––––, The compactness of the riemann manifold of an abstract field of algebraic functions, Bull. Amer. Math. Soc. 50 (1944), 683–691.
  • ––––, Reduction of the singularities of algebraic three dimensional varieties, Ann. Math. 45 (1944), 472–542.
  • Oscar Zariski and Pierre Samuel, Commutative algebra II, Springer, New York, 1960.