Functorial desingularization over Q: boundaries and the embedded case

Article
  • 3 Downloads

Abstract

Our main result establishes functorial desingularization of noetherian quasi-excellent schemes over Q with ordered boundaries. A functorial embedded desingularization of quasi-excellent schemes of characteristic zero is deduced. Furthermore, a standard simple argument extends these results to other categories including, in particular, (equivariant) embedded desingularization of the following objects of characteristic zero: qe algebraic stacks, qe formal schemes, complex and non-archimedean analytic spaces. We also obtain a semistable reduction theorem for formal schemes.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [Ber15]
    V. Berkovich, Complex analytic vanishing cycles for formal schemes, http://www.wisdom.weizmann.ac.il/~vova/FormIV_2015.pdf, 2015.Google Scholar
  2. [BM97]
    E. Bierstone and P. D. Milman, Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant, Inventiones Mathematicae 128 (1997), 207–302.Google Scholar
  3. [BM08]
    E. Bierstone and P. D. Milman, Functoriality in resolution of singularities, Kyoto University. Research Institute for Mathematical Sciences. Publications 44 (2008), 609–639.Google Scholar
  4. [BMT11]
    E. Bierstone, P. D. Milman and M. Temkin, Q-universal desingularization, Asian Journal of Mathematics 15 (2011), 229–249.Google Scholar
  5. [CJS13]
    V. Cossart, U. Jannsen and S. Saito, Canonical embedded and non-embedded resolution of singularities for excellent two-dimensional schemes, preprint, http://arxiv.org/abs/0905.2191 (2013).Google Scholar
  6. [Con07]
    B. Conrad, Deligne’s notes on Nagata compactifications, Journal of the Ramanujan Mathematical Society 22 (2007), 205–257.Google Scholar
  7. [Gro67]
    A. Grothendieck, Éléments de géométrie algébrique. I–IV, Institut des Hautes Études Scientifiques. Publications Mathématiques 4, 8, 11, 17, 20, 24, 28, 32 (1960–1967).Google Scholar
  8. [Hir64]
    H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II, Annals of Mathematics 79 (1964), 109–203; 79 (1964), 205–326.Google Scholar
  9. [IT14]
    L. Illusie and M. Temkin, Exposé VIII. Gabber’s modification theorem (absolute case), Astérisque 364 (2014), 103–160Google Scholar
  10. [Kat94]
    K. Kato, Toric singularities, American Journal of Mathematics 116 (1994), 1073–1099.Google Scholar
  11. [KKMSD73]
    G. Kempf, F. F. Knudsen, D. Mumford and B. Saint-Donat, Toroidal Embeddings. I, Lecture Notes in Mathematics, Vol. 339, Springer-Verlag, Berlin–New York, 1973.Google Scholar
  12. [Kol07]
    J. Kollár, Lectures on Resolution of Singularities, Annals of Mathematics Studies, Vol. 166, Princeton University Press, Princeton, NJ, 2007.Google Scholar
  13. [Kol08]
    J. Kollár, Semi log resolution, preprint, http://arxiv.org/abs/0812.3592 (2008).Google Scholar
  14. [Niz06]
    W. Niziol, Toric singularities: log-blow-ups and global resolutions, Journal of Algebraic Geometry 15 (2006), 1–29.Google Scholar
  15. [Tem08]
    M. Temkin, Desingularization of quasi-excellent schemes in characteristic zero, Advances in Mathematics 219 (2008), 488–522.Google Scholar
  16. [Tem12]
    M. Temkin, Functorial desingularization of quasi-excellent schemes in characteristic zero: the nonembedded case, Duke Mathematical Journal 161 (2012), 2207–2254.Google Scholar
  17. [Tru12]
    D. Trushin, Algebraization of a Cartier divisor, preprint, http://arxiv.org/abs/1210.4176 (2012).Google Scholar
  18. [Wło05]
    J. Włodarczyk, Simple Hironaka resolution in characteristic zero, Journal of the American Mathematical Society 18 (2005), 779–822.Google Scholar

Copyright information

© Hebrew University of Jerusalem 2018

Authors and Affiliations

  1. 1.Einstein Institute of MathematicsThe Hebrew University of JerusalemGiv’at Ram, JerusalemIsrael

Personalised recommendations