Desingularization of toric and binomial varieties
Authors:
Edward Bierstone and Pierre D. Milman
Journal:
J. Algebraic Geom. 15 (2006), 443-486
DOI:
https://doi.org/10.1090/S1056-3911-06-00430-9
Published electronically:
March 1, 2006
MathSciNet review:
2219845
Full-text PDF
Abstract |
References |
Additional Information
Abstract: We give a combinatorial algorithm for equivariant embedded resolution of singularities of a toric variety defined over a perfect field. The algorithm is realized by a finite succession of blowings-up with smooth invariant centres that satisfy the normal flatness condition of Hironaka. The results extend to more general varieties defined locally by binomial equations.
- Bruce Michael Bennett, On the characteristic functions of a local ring, Ann. of Math. (2) 91 (1970), 25–87. MR 252388, DOI https://doi.org/10.2307/1970601
- Edward Bierstone and Pierre D. Milman, Semianalytic and subanalytic sets, Inst. Hautes Études Sci. Publ. Math. 67 (1988), 5–42. MR 972342
- Edward Bierstone and Pierre D. Milman, Uniformization of analytic spaces, J. Amer. Math. Soc. 2 (1989), no. 4, 801–836. MR 1001853, DOI https://doi.org/10.1090/S0894-0347-1989-1001853-2
- Edward Bierstone and Pierre D. Milman, Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant, Invent. Math. 128 (1997), no. 2, 207–302. MR 1440306, DOI https://doi.org/10.1007/s002220050141
- Edward Bierstone and Pierre D. Milman, Standard basis along a Samuel stratum, and implicit differentiation, The Arnoldfest (Toronto, ON, 1997) Fields Inst. Commun., vol. 24, Amer. Math. Soc., Providence, RI, 1999, pp. 81–113. MR 1733569
- Edward Bierstone and Pierre D. Milman, Desingularization algorithms. I. Role of exceptional divisors, Mosc. Math. J. 3 (2003), no. 3, 751–805, 1197 (English, with English and Russian summaries). {Dedicated to Vladimir Igorevich Arnold on the occasion of his 65th birthday}. MR 2078560, DOI https://doi.org/10.17323/1609-4514-2003-3-3-751-805
- David A. Cox, Toric varieties and toric resolutions, Resolution of singularities (Obergurgl, 1997) Progr. Math., vol. 181, Birkhäuser, Basel, 2000, pp. 259–284. MR 1748623
[DP]DP C. De Concini and C. Procesi, Complete symmetric varieties II, Algebraic Groups and Related Topics, ed. R. Hotta, Adv. Studies in Pure Math., vol. 6, Kinokuniya, Tokyo and North-Holland, Amsterdam, New York, Oxford, 1985, pp. 481–513.
- David Eisenbud and Bernd Sturmfels, Binomial ideals, Duke Math. J. 84 (1996), no. 1, 1–45. MR 1394747, DOI https://doi.org/10.1215/S0012-7094-96-08401-X
- Santiago Encinas and Orlando Villamayor, A new proof of desingularization over fields of characteristic zero, Proceedings of the International Conference on Algebraic Geometry and Singularities (Spanish) (Sevilla, 2001), 2003, pp. 339–353. MR 2023188, DOI https://doi.org/10.4171/RMI/350
- William Fulton, Introduction to toric varieties, Annals of Mathematics Studies, vol. 131, Princeton University Press, Princeton, NJ, 1993. The William H. Roever Lectures in Geometry. MR 1234037
- I. M. Gel′fand, M. M. Kapranov, and A. V. Zelevinsky, Discriminants, resultants, and multidimensional determinants, Mathematics: Theory & Applications, Birkhäuser Boston, Inc., Boston, MA, 1994. MR 1264417
- Pedro Daniel González Pérez and Bernard Teissier, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris 334 (2002), no. 5, 379–382 (English, with English and French summaries). MR 1892938, DOI https://doi.org/10.1016/S1631-073X%2802%2902273-2
- Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. MR 0463157
- Heisuke Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II, Ann. of Math. (2) 79 (1964), 109–203; ibid. (2) 79 (1964), 205–326. MR 0199184, DOI https://doi.org/10.2307/1970547
- Heisuke Hironaka, Idealistic exponents of singularity, Algebraic geometry (J. J. Sylvester Sympos., Johns Hopkins Univ., Baltimore, Md., 1976) Johns Hopkins Univ. Press, Baltimore, Md., 1977, pp. 52–125. MR 0498562
- G. Kempf, Finn Faye Knudsen, D. Mumford, and B. Saint-Donat, Toroidal embeddings. I, Lecture Notes in Mathematics, Vol. 339, Springer-Verlag, Berlin-New York, 1973. MR 0335518
- R. Narasimhan, Hyperplanarity of the equimultiple locus, Proc. Amer. Math. Soc. 87 (1983), no. 3, 403–408. MR 684627, DOI https://doi.org/10.1090/S0002-9939-1983-0684627-8
- Tadao Oda, Torus embeddings and applications, Tata Institute of Fundamental Research Lectures on Mathematics and Physics, vol. 57, Tata Institute of Fundamental Research, Bombay; by Springer-Verlag, Berlin-New York, 1978. Based on joint work with Katsuya Miyake. MR 546291
- Tadao Oda, Convex bodies and algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 15, Springer-Verlag, Berlin, 1988. An introduction to the theory of toric varieties; Translated from the Japanese. MR 922894
- Bernd Sturmfels, Gröbner bases of toric varieties, Tohoku Math. J. (2) 43 (1991), no. 2, 249–261. MR 1104431, DOI https://doi.org/10.2748/tmj/1178227496
- Bernd Sturmfels, Gröbner bases and convex polytopes, University Lecture Series, vol. 8, American Mathematical Society, Providence, RI, 1996. MR 1363949
- Bernard Teissier, Valuations, deformations, and toric geometry, Valuation theory and its applications, Vol. II (Saskatoon, SK, 1999) Fields Inst. Commun., vol. 33, Amer. Math. Soc., Providence, RI, 2003, pp. 361–459. MR 2018565
- O. E. Villamayor U., Patching local uniformizations, Ann. Sci. École Norm. Sup. (4) 25 (1992), no. 6, 629–677. MR 1198092
[W]W J. Włodarczyk, Simple Hironaka resolution in characteristic zero, preprint, 2004, math.AG/0401401.
- Oscar Zariski and Pierre Samuel, Commutative algebra, Volume I, The University Series in Higher Mathematics, D. Van Nostrand Company, Inc., Princeton, New Jersey, 1958. With the cooperation of I. S. Cohen. MR 0090581
[Be]Be B.M. Bennett, On the characteristic function of a local ring, Ann. of Math. (2) 91 (1970), 25–87.
[BM1]BMihes E. Bierstone and P.D. Milman, Semianalytic and subanalytic sets, Inst. Hautes Études Sci. Publ. Math. 67 (1988), 5–42.
[BM2]BMjams E. Bierstone and P.D. Milman, Uniformization of analytic spaces, J. Amer. Math. Soc. 2 (1989), 801–836.
[BM3]BMinv E. Bierstone and P.D. Milman, Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant, Invent. Math. 128 (1997), 207–302.
[BM4]BMsam E. Bierstone and P.D. Milman, Standard basis along a Samuel stratum and implicit differentiation, The Arnoldfest, Proceedings of a Conference in Honour of V.I. Arnold for his Sixtieth Birthday, ed. E. Bierstone, B. Khesin, A. Khovanskii and J. Marsden, Fields Inst. Comm., vol. 24, Amer. Math. Soc., Providence, 1999, pp. 81–113.
[BM5]BMda1 E. Bierstone and P.D. Milman, Desingularization algorithms I. Role of exceptional divisors, Moscow Math. J. 3 (2003), 751–805.
[C]Cox D.A. Cox, Toric varieties and toric resolution, Resolution of Singularities, A Research Textbook in Tribute to Oscar Zariski, Progress in Math., vol. 181, Birkhäuser, Basel, Boston, Berlin, 2000, pp. 259–284.
[DP]DP C. De Concini and C. Procesi, Complete symmetric varieties II, Algebraic Groups and Related Topics, ed. R. Hotta, Adv. Studies in Pure Math., vol. 6, Kinokuniya, Tokyo and North-Holland, Amsterdam, New York, Oxford, 1985, pp. 481–513.
[ES]ES D. Eisenbud and B. Sturmfels, Binomial ideals, Duke Math. J. 84 (1996), 1–45.
[EnV]EnV S. Encinas and O. Villamayor, A new proof of desingularization over fields of characteristic zero, Proceedings of the International Conference on Algebraic Geometry and Singularities, Sevilla, 2001, Rev. Mat. Iberoamericana 19 (2003), 339–353.
[F]Ful W. Fulton, Introduction to toric varieties, Ann. of Math. Studies, vol. 131, Princeton Univ. Press, Princeton, 1993.
[GKK]GKK I. M. Gelfand, M. M. Kapranov and A. V. Zelevinsky, Discriminants, Resultants and Multidimensional Determinants, Birkhäuser, Boston, Basel, Berlin, 1994.
[GoT]GoT P.D. González Pérez and B. Teissier, Embedded resolutions of non necessarily normal affine toric varieties, C.R. Acad. Sci. Paris, Ser. I 334 (2002), 379–382.
[Ha]Hart R. Hartshorne, Algebraic geometry, Grad. Texts in Math., vol. 52, Springer, New York, 1977.
[Hi1]Hann H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero: I, II, Ann. of Math. (2) 79 (1964), 109–326.
[Hi2]Hid H. Hironaka, Idealistic exponents of singularity, Algebraic geometry, J.J. Sylvester Sympos., Johns Hopkins Univ., Baltimore 1976, Johns Hopkins Univ. Press, Baltimore, 1977, pp. 52–125.
[KKMS]KKMS G. Kempf, K. Knudson, D. Mumford and B. Saint-Donat, Toroidal embeddings I, Lecture Notes in Math. vol. 339, Springer-Verlag, Berlin, Heidelberg, New York, 1973.
[N]Nar R. Narasimhan, Hyperplanarity of the equimultiple locus, Proc. Amer. Math. Soc. 87 (1983), 403–406.
[O1]Oda1 T. Oda, Lectures on torus embeddings and applications, Tata Inst. Fund. Research 58, Springer-Verlag, Berlin, Heidelberg, New York, 1978.
[O2]Oda T. Oda, Convex Bodies and Algebraic Geometry, An Introduction to the Theory of Toric Varieties, Springer-Verlag, Berlin, Heidelberg, New York, 1988.
[S1]Stu1 B. Sturmfels, Gröbner bases of toric varieties, Tôhoku Math. J. 43 (1991), 249–261.
[S2]Stu2 B. Sturmfels, Gröbner bases and convex polytopes, University Lecture Series, vol. 8, Amer. Math. Soc., Providence, 1996.
[T]Tei B. Teissier, Valuations, deformations, and toric geometry, Valuation theory and its applications, Vol. II, ed. F.-V. Kuhlmann, S. Kuhlmann and M. Marshall, Fields Inst. Comm., vol. 33, Amer. Math. Soc., Providence, 2003, pp. 361–459.
[V]V O. Villamayor, Patching local uniformizations, Ann. Sci. Ecole Norm Sup. Paris (4) 25 (1992), 629–677.
[W]W J. Włodarczyk, Simple Hironaka resolution in characteristic zero, preprint, 2004, math.AG/0401401.
[ZS]ZS O. Zariski and P. Samuel, Commutative Algebra Vol. I, Van Nostrand, Princeton, 1958.
Additional Information
Edward Bierstone
Affiliation:
Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 2E4
Email:
bierston@math.toronto.edu
Pierre D. Milman
Affiliation:
Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 2E4
Email:
milman@math.toronto.edu
Received by editor(s):
December 3, 2004
Received by editor(s) in revised form:
September 15, 2005
Published electronically:
March 1, 2006
Additional Notes:
The authors’ research was supported in part by NSERC grants OGP0009070 and OGP0008949