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On semistable degenerations of Fano varieties

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Abstract

Consider a family of Fano varieties \(\pi :X \rightarrow B\ni o\) over a curve germ with a smooth total space X. Assume that the generic fiber is smooth and the special fiber \(F=\pi ^{-1}(o)\) has simple normal crossings. Then F is called a semistable degeneration of Fano varieties. We show that the dual complex of F is a simplex of dimension \(\hbox {\,\char 054\,}\mathrm {dim}\, F\). Simplices of any admissible dimension can be realized for any dimension of the fiber. Using this result and the Minimal Model Program in dimension 3 we reproduce the classification of the semistable degenerations of del Pezzo surfaces obtained by Fujita. We also show that the maximal degeneration is unique and has trivial monodromy in dimension \(\hbox {\,\char 054\,}\,3\).

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Acknowledgements

The author would like to thank Lev Soukhanov for the conversations that inspired the present paper, Yuri Prokhorov for constant support and helpful advice, Igor Krylov and the Korean Institute for Advanced Study for hospitality during the work on this paper, Dmitry Mineyev and Costya Shramov for useful discussions.

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  1. Steklov Mathematical Institute of Russian Academy of Sciences, 8 Gubkina str., Moscow, 119991, Russia

    Konstantin Loginov

  2. Laboratory of Algebraic Geometry, National Research University Higher School of Economics, 6 Usacheva str., Moscow, 119048, Russia

    Konstantin Loginov

  3. Laboratory of AGHA, Moscow Institute of Physics and Technology, 9 Institutskiy per., Dolgoprudny, Moscow Region, 141701, Russia

    Konstantin Loginov

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Correspondence to Konstantin Loginov.

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The author was partially supported by the HSE University Basic Research Program, Russian Academic Excellence Project “5-100”, Foundation for the Advancement of Theoretical Physics and Mathematics “BASIS”, and the Simons Foundation.

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Loginov, K. On semistable degenerations of Fano varieties. European Journal of Mathematics 8, 991–1005 (2022). https://doi.org/10.1007/s40879-021-00457-w

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  • DOI: https://doi.org/10.1007/s40879-021-00457-w

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