Cycles of singularities appearing in the resolution problem in positive characteristic
Authors:
Herwig Hauser and Stefan Perlega
Journal:
J. Algebraic Geom. 28 (2019), 391-403
DOI:
https://doi.org/10.1090/jag/718
Published electronically:
January 4, 2019
MathSciNet review:
3912062
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Abstract |
References |
Additional Information
Abstract: We present a hypersurface singularity in positive characteristic which is defined by a purely inseparable power series, and a sequence of point blowups so that, after applying the blowups to the singularity, the same type of singularity reappears after the last blowup, with just certain exponents of the defining power series shifted upwards. The construction hence yields a cycle. Iterating this cycle leads to an infinite increase of the residual order of the defining power series. This disproves a theorem claimed by Moh about the stability of the residual order under sequences of blowups. It is not a counterexample to the resolution in positive characteristic since larger centers are also permissible and prevent the phenomenon from happening.
References
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References
- Shreeram Shankar Abhyankar, Nonsplitting of valuations in extensions of two dimensional regular local domains, Math. Ann. 170 (1967), 87–144. MR 0207698
- 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
- Angélica Benito and Orlando E. Villamayor U., Monoidal transforms and invariants of singularities in positive characteristic, Compos. Math. 149 (2013), no. 8, 1267–1311. MR 3103065
- Vincent Cossart and Olivier Piltant, Resolution of singularities of threefolds in positive characteristic. I. Reduction to local uniformization on Artin-Schreier and purely inseparable coverings, J. Algebra 320 (2008), no. 3, 1051–1082. MR 2427629
- Vincent Cossart and Olivier Piltant, Resolution of singularities of threefolds in positive characteristic. II, J. Algebra 321 (2009), no. 7, 1836–1976. MR 2494751
- Steven Dale Cutkosky, A skeleton key to Abhyankar’s proof of embedded resolution of characteristic p surfaces, Asian J. Math. 15 (2011), no. 3, 369–416. MR 2838213
- Santiago Encinas and Herwig Hauser, Strong resolution of singularities in characteristic zero, Comment. Math. Helv. 77 (2002), no. 4, 821–845. MR 1949115
- Herwig Hauser, On the problem of resolution of singularities in positive characteristic (or: a proof we are still waiting for), Bull. Amer. Math. Soc. (N.S.) 47 (2010), no. 1, 1–30. MR 2566444
- 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
- Heisuke Hironaka, Resolution of singularities, manuscript distributed at the CMI Summer School, 2012, 138 pp.
- H. Hauser and S. Perlega, A new proof for the embedded resolution of surface singularities, manuscript, 2018.
- H. Hauser and S. Perlega, Characterizing the increase of the residual order under blowup in positive characteristic, Publ. RIMS (2019), to appear.
- Herwig Hauser and Dominique Wagner, Alternative invariants for the embedded resolution of purely inseparable surface singularities, Enseign. Math. 60 (2014), no. 1-2, 177–224. MR 3262439
- Hiraku Kawanoue and Kenji Matsuki, Resolution of singularities of an idealistic filtration in dimension 3 after Benito-Villamayor, Minimal models and extremal rays (Kyoto, 2011) Adv. Stud. Pure Math., vol. 70, Math. Soc. Japan, [Tokyo], 2016, pp. 115–214. MR 3617780
- T. T. Moh, On a stability theorem for local uniformization in characteristic $p$, Publ. Res. Inst. Math. Sci. 23 (1987), no. 6, 965–973. MR 935710
- Orlando Villamayor, Constructiveness of Hironaka’s resolution, Ann. Sci. École Norm. Sup. (4) 22 (1989), no. 1, 1–32. MR 985852
- O. E. Villamayor U., Patching local uniformizations, Ann. Sci. École Norm. Sup. (4) 25 (1992), no. 6, 629–677. MR 1198092
- Jarosław Włodarczyk, Simple Hironaka resolution in characteristic zero, J. Amer. Math. Soc. 18 (2005), no. 4, 779–822. MR 2163383
Additional Information
Herwig Hauser
Affiliation:
Faculty of Mathematics, University of Vienna, A-1090 Vienna, Austria
MR Author ID:
82620
Email:
herwig.hauser@univie.ac.at
Stefan Perlega
Affiliation:
Faculty of Mathematics, University of Vienna, A-1090 Vienna, Austria
MR Author ID:
945358
Email:
stefan.perlega@univie.ac.at
Received by editor(s):
August 23, 2017
Received by editor(s) in revised form:
October 7, 2017, November 2, 2017, November 21, 2017, December 12, 2017, and January 1, 2018
Published electronically:
January 4, 2019
Additional Notes:
Supported by project P-25652 of the Austrian Science Fund FWF
Article copyright:
© Copyright 2019
University Press, Inc.