Toric geometry and the Semple–Nash modification

Original Paper


This paper proposes some material towards a theory of general toric varieties without the assumption of normality. Their combinatorial description involves a fan to which is attached a set of semigroups subjected to gluing-up conditions. In particular it contains a combinatorial construction of the blowing up of a sheaf of monomial ideals on a toric variety. In the second part this is used to show that iterating the Semple–Nash modification or its characteristic-free avatar provides a local uniformization of any monomial valuation of maximal rank dominating a point of a toric variety.


Toric geometry Semple–Nash modification Logarithmic Jacobian ideal 

Mathematics Subject Classification (2000)

14M25 14E15 14B05 



We are grateful to Monique Lejeune-Jalabert for introducing us to the reference [27], to Ezra Miller for bringing to our attention the work of Howard M. Thompson, to Michael Thaddeus for detecting errors in previous versions of this work. We also mention that while this paper was in preparation, Mr. Daniel Duarte provided in [7] a proof of a result similar to our main result, a priori somewhat stronger and giving an effective bound on the number of steps required for desingularization, in the two-dimensional case. A preprint of Dima Grigoriev and Pierre Milman (see [17]) also provided an approach to the Semple–Nash desingularization problem for toric varieties with explicit results in dimension 2. In particular it contains a version of our Lemma 70. We thank the referee for his careful work and helpful suggestions. We also thank Patrice Philippon for his help, David Cox for his remarks and the Institut Mathématique de Jussieu and the Dpto. Álgebra, Universidad Complutense de Madrid for their hospitality.


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Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Instituto de Ciencias Matemáticas-CSIC-UAM-UC3M-UCM. Dpto. de Algebra. Facultad de Ciencias MatemáticasUniversidad Complutense de MadridMadridSpain
  2. 2.Institut Mathématique de Jussieu, UMR 7586 du CNRSParisFrance

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