## Proximity relations for real rank one valuations dominating a local regular ring.(English)Zbl 1094.13535

Summary: We study 0-dimensional real rank one valuations centered in a regular local ring of dimension $$n\geq 2$$ such that the associated valuation ring can be obtained from the regular ring by a sequence of quadratic transforms. We define two classical invariants associated to the valuation (the refined proximity matrix and the multiplicity sequence) and we show that are equivalent data of the valuation.

### MSC:

 13F30 Valuation rings 13H05 Regular local rings
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### References:

 [1] Abhyankar, S. S.: On the valuations centered in a local domain. Amer. J. Math. 78 (1956), 321-348. · Zbl 0074.26301 [2] Abhyankar, S. S.: Desingularization of plane curves. In Singularities, Part 1 (Arcata, Calif., 1981), 1-45. Proc. Sympos. Pure Math. 40. Amer. Math. Soc., Providence, RI, 1983. · Zbl 0521.14005 [3] Abhyankar, S. S.: Good points of a hypersurface. Adv. in Math. 68 (1988), 87-256. · Zbl 0657.14008 [4] Abhyankar, S. S.: Algebraic Geometry for Scientists and Engineers. Mathematical Surveys and Monographs 35. American Mathematical So- ciety, Providence, RI, 1990. · Zbl 0709.14001 [5] Abhyankar, S. S.: Ramification Theoretic Methods in Algebraic Geome- try. Annals of Mathematics Studies 43. Princeton University Press, Prince- ton, N.J. 1959. · Zbl 0101.38201 [6] Abhyankar, S. S.: Resolution of Singularities of Embedded Algebraic Sur- faces. Springer-Verlag. New York Inc., 1998. · Zbl 0914.14006 [7] Abhyankar, S. S.: Resolution of singularities and modular Galois theory. Bull. Amer. Math. Soc. (N.S.) 38 (2001), no. 2, 131-169. · Zbl 0999.12003 [8] Aparicio, J., Granja, A. and Sánchez-Giralda, T.: On proximity relations for valuations dominating a two-dimensional regular local ring. Rev. Mat. Iberoamericana 15 (1999), no. 3, 621-634. · Zbl 0968.13011 [9] Delgado, D., Galindo, C. and Núñez, A.: Saturation for valuations on two-dimensional regular rings. Math. Z. 234 (2000), 519-550. · Zbl 0967.13019 [10] Granja, A. and Sánchez-Giralda, T.: Enriques graphs of plane curves. Comm. Algebra 20 (1992), no. 2, 527-562. A. Granja and C. Rodríguez · Zbl 0752.14020 [11] Lipman, J.: Proximity inequalities for complete ideals in two-dimensional regular local rings. In Commutative algebra: syzygies, multiplicities, and bi- rational algebra (South Hadley, MA, 1992), 293-306. Contemp. Math. 159. Amer. Math. Soc., Providence, RI, 1994. · Zbl 0814.13016 [12] Nagata, M.: Local Rings. Interscience Tracts in Pure and Applied Math- ematics 13, Interscience Publishers, a division of John Wiley & Sons, New York-London, 1962. [13] Shannon, D. L.: Monoidal transforms of regular rings. Amer. J. Math. 95 (1973), 294-320. · Zbl 0271.14003 [14] Vaquié, M.: Valuations. In Resolution of singularities (Obergurgl, 1997), 539-590. Progr. Math. 181, Birkhäuser, Basel, 2000. [15] Zariski, O.: Local uniformization on algebraic varieties. Ann. of Math. 41 (1940), 852-896. · Zbl 0025.21601 [16] Zariski, O. and Samuel, P.: Commutative Algebra, Vols. I and II. D. Van Nostrand Company, Inc., Princeton, New Jersey, 1958, 1960.
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