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Some elementary examples of non-liftable varieties

Authors: Piotr Achinger and Maciej Zdanowicz
Journal: Proc. Amer. Math. Soc. 145 (2017), 4717-4729
MSC (2010): Primary 14D15; Secondary 14G17
Published electronically: June 5, 2017
MathSciNet review: 3691989
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Abstract: We present some simple examples of smooth projective varieties in positive characteristic, arising from linear algebra, which do not admit a lifting neither to characteristic zero, nor to the ring of Witt vectors of length $2$. Our first construction is the blow-up of the graph of the Frobenius morphism of a homogeneous space. The second example is a blow-up of $\mathbb {P}^3$ in a ‘purely characteristic-$p$’ configuration of points and lines.

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Piotr Achinger
Affiliation: Banach Center, Instytut Matematyczny PAN, Śniadeckich 8, Warsaw, Poland

Maciej Zdanowicz
Affiliation: Wydział Matematyki, Informatyki i Mechaniki UW, Banacha 2, Warsaw, Poland

Keywords: mod $p^2$ deformation, Frobenius morphism, characteristic $p$ line configurations
Received by editor(s): July 5, 2016
Received by editor(s) in revised form: December 9, 2016, and December 14, 2016
Published electronically: June 5, 2017
Additional Notes: The first author was supported by NCN OPUS grant number UMO-2015/17/B/ST1/02634
The second author was supported by NCN PRELUDIUM grant number UMO-2014/13/N/ST1/02673. This work was partially supported by the grant 346300 for IMPAN from the Simons Foundation and the matching 2015–2019 Polish MNiSW fund
Communicated by: Lev Borisov
Article copyright: © Copyright 2017 American Mathematical Society