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Some examples of non-smoothable Gorenstein Fano toric threefolds

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Abstract

We present a combinatorial criterion on reflexive polytopes of dimension 3 which gives a local-to-global obstruction for the smoothability of the corresponding Fano toric threefolds. As a result, we show an example of a singular Gorenstein Fano toric threefold which has compound Du Val, hence smoothable, singularities but is not smoothable.

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References

  1. Akhtar, M., Coates, T., Corti, A., Heuberger, L., Kasprzyk, A., Oneto, A., Petracci, A., Prince, T., Tveiten, K.: Mirror symmetry and the classification of orbifold del Pezzo surfaces. Proc. Am. Math. Soc. 144(2), 513–527 (2016)

    MathSciNet  MATH  Google Scholar 

  2. Akhtar, M., Coates, T., Galkin, S., Kasprzyk, A.M.: Minkowski polynomials and mutations. SIGMA Symmetry Integrability. Geom. Methods Appl. 8, Paper 094, 17 (2012)

  3. Coates, T., Corti, A., Galkin, S., Golyshev, V., Kasprzyk, A.: Mirror symmetry and Fano manifolds. In: European Congress of Mathematics, pp. 285–300 . Eur. Math. Soc., Zürich (2013)

  4. Cox, D.A., Little, J.B., Schenck, H.K.: Toric Varieties, Graduate Studies in Mathematics, vol. 124. American Mathematical Society, Providence (2011)

    Google Scholar 

  5. Ilten, N.O.: Mutations of Laurent polynomials and flat families with toric fibers. SIGMA Symmetry Integrability Geom. Methods Appl. 8(047), 7 (2012)

    MathSciNet  MATH  Google Scholar 

  6. Kreuzer, M., Skarke, H.: Classification of reflexive polyhedra in three dimensions. Adv. Theor. Math. Phys. 2(4), 853–871 (1998)

    MathSciNet  MATH  Google Scholar 

  7. Manetti, M.: Differential graded Lie algebras and formal deformation theory. In: Algebraic geometry—Seattle 2005. Part 2, Proc. Sympos. Pure Math., vol. 80, pp. 785–810. American Mathematical Society, Providence (2009)

  8. Namikawa, Y.: Smoothing Fano \(3\)-folds. J. Algebraic Geom. 6(2), 307–324 (1997)

    MathSciNet  MATH  Google Scholar 

  9. Petracci, A.: On Mirror Symmetry for Fano varieties and for singularities. Ph.D. thesis, Imperial College London (2017)

  10. Prokhorov, Y.G.: The degree of Fano threefolds with canonical Gorenstein singularities. Mat. Sb. 196(1), 81–122 (2005)

    MathSciNet  MATH  Google Scholar 

  11. Sernesi, E.: Deformations of Algebraic Schemes, Grundlehren der Mathematischen Wissenschaften, vol. 334. Springer, Berlin (2006)

    MATH  Google Scholar 

  12. Totaro, B.: Jumping of the nef cone for Fano varieties. J. Algebraic Geom. 21(2), 375–396 (2012)

    MathSciNet  MATH  Google Scholar 

  13. Vistoli, A.: The deformation theory of local complete intersections ArXiv preprint arXiv:alg-geom/9703008

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Acknowledgements

The results in this note have appeared in my Ph.D. thesis [9], which was supervised by Alessio Corti; I would like to thank him for suggesting this problem to me and for sharing his ideas. I am grateful to Victor Przyjalkowski for bringing Prokhorov’s conjecture to my attention. The author was funded by Tom Coates’ ERC Consolidator Grant 682603 and by Alexander Kasprzyk’s EPSRC Fellowship EP/N022513/1.

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  1. Institut für Mathematik, Freie Universität Berlin, Arnimallee 3, 14195, Berlin, Germany

    Andrea Petracci

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  1. Andrea Petracci
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Correspondence to Andrea Petracci.

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Petracci, A. Some examples of non-smoothable Gorenstein Fano toric threefolds. Math. Z. 295, 751–760 (2020). https://doi.org/10.1007/s00209-019-02369-8

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