Abstract
In this paper, we show that all the nontrivial valuations on surfaces can be given by the infinite sequences of blowing-ups, and give the process of blowing-ups.
Similar content being viewed by others
References
Roquette, P.: History of valuation theory Part I. Fields Institute communication series. International Conference and Workshop on Valuation Theory, July 26–August 11, 1999, http://2002-rzuser.uni-heidelberg.de
Green, B.: Automorphisms of formal power series rings over a valuation ring. Fields Institute Communications, International Conference and Workshop on Valuation Theory, July 26–August 11, 1999
Herrera, F. J., Olalla, M. A., Vicente, J. L.: Valuations in fields of power series. Rev. Mat. Iberoamericana, 19, 467–482 (2003)
Olalla, M. A.: On the dimension of discrtete valuations of k((X 1, ..., X n)). Comm. Algera, 333, 27–32 (2001)
Herrera, F. J., Olalla, M. A., Vicente, J. L.: Rank one discrete valuations of k((X 1, ..., X n)). Comm. Algebra, 35, 2533–2551 (2007)
Hartshorne, R.: Algebraic Geometry (GTM52), Springer-Verlag, New York, 1977
Xu, N.: Classification of valuations on surfaces. Journal of GSCAS, 26(3), 289–295 (2009)
Mo, Z. J., Lan, Y. Z., Zhao, C. L.: Algebra 2 (in Chinese), Peking University Press, Beijing, 2001
Endler, O.: Valuation Theory, Springer-Verlag, Berlin, New York, 1972
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Xu, N. Blowing-ups and valuations on surfaces. Acta. Math. Sin.-English Ser. 27, 1305–1314 (2011). https://doi.org/10.1007/s10114-011-8670-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10114-011-8670-5