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Universal stacky semistable reduction

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Abstract

Given a log smooth morphism f: X → S of toroidal embeddings, we perform a Raynaud-Gruson type operation on f to make it flat and with reduced fibers. We do this by studying the geometry of the associated map of cone complexes C(X) → C(S). As a consequence, we show that the toroidal part of semistable reduction of Abramovich-Karu can be done in a canonical way.

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References

  1. D. Abramovich, Q. Chen, S. Marcus, M. Ulirsch and J. Wise, Skeletons and fans of logarithmic structures, in Nonarchimedean and Tropical Geometry, Simons Symposia, Springer, Cham, 2016, pp. 287–336.

    Chapter  Google Scholar 

  2. D. Abramovich and K. Karu, Weak semistable reduction in characteristic 0, Inventiones Mathematicae 139 (2000), 241–273.

    Article  MathSciNet  Google Scholar 

  3. K. Adiprasito, G. Liu and M. Temkin, Semistable reduction in characteristic 0, Séminaire Lotharingien de Combinatoire 82B (2020), Article no. 25.

  4. K. Ascher and S. Molcho, Logarithmic stable toric varieties and their moduli, Algebraic Geometry 3 (2016), 296–319.

    Article  MathSciNet  Google Scholar 

  5. N. Borne and A. Vistoli, Parabolic sheaves on logarithmic schemes, Advances in Mathematics 231 (2012), 1327–1363.

    Article  MathSciNet  Google Scholar 

  6. W. D. Gillam and S. Molcho, A theory of stacky fans, https://arxiv.org/abs/1512.07586.

  7. K. Kato, Logarithmic Structures of Fontaine-Illusie, in Algebraic Analysis, Geometry, and Number Theory (Baltimore, MD, 1988), Johns Hopkins University Press, Baltimore, MD, 1989, pp. 191–224.

    Google Scholar 

  8. K. Kato, Toric singularities, American Journal of Mathematics 116 (1994), 1073–1099.

    Article  MathSciNet  Google Scholar 

  9. G. Kempf, F. F. Knudsen, D. Mumford and B. Saint-Donat, Toroidal Embeddings. I, Lecture Notes in Mathematics, Vol. 339, Springer, Berlin-New York, 1973.

    Book  Google Scholar 

  10. M. M. Kapranov, B. Sturmfels and A. V. Zelevinsky, Quotients of toric varieties, Mathematische Annalen 290 (1991), 643–655.

    Article  MathSciNet  Google Scholar 

  11. A. Ogus, Lectures on Logarithmic Algebraic Geometry, Cambridge Studies in Advanced Mathematics, Vol. 178, Cambridge University Press, Cambridge, 2018.

    Book  Google Scholar 

  12. M. Olsson, A stacky semistable reduction theorem, International Mathematics Research Notices (2004), 1497–1509.

  13. T. Tsuji, Saturated morphisms of logarithmic schemes, Tunisian Journal of Mathematics 1 (2019), 185–220.

    Article  MathSciNet  Google Scholar 

  14. I. Tyomkin, Tropical geometry and correspondence theorems via toric stacks, Mathematische Annalen 353 (2012), 945–995.

    Article  MathSciNet  Google Scholar 

  15. J. Wise, Moduli of morphisms of logarithmic schemes, Algebra & Number Theory 10 (2016), 695–735.

    Article  MathSciNet  Google Scholar 

Download references

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Authors and Affiliations

  1. Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Givat Ram, Jerusalem, 91904, Israel

    Sam Molcho

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  1. Sam Molcho
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Correspondence to Sam Molcho.

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Molcho, S. Universal stacky semistable reduction. Isr. J. Math. 242, 55–82 (2021). https://doi.org/10.1007/s11856-021-2118-0

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  • DOI: https://doi.org/10.1007/s11856-021-2118-0

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