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Mosc. Math. J., 2017, Volume 17, Number 1, Pages 15–33 (Mi mmj623)  

This article is cited in 2 scientific papers (total in 2 papers)

Remarks on Mukai threefolds admitting $\mathbb C^*$ action

Sławomir Dinewa, Grzegorz Kapustkaba, Michał Kapustkac

a Department of Mathematics and Computer Science, Jagiellonian University, Kraków, Poland
b Institute of Mathematics of the Polish Academy of Sciences, Warsaw
c University of Stavanger, Norway

Abstract: We investigate geometric properties of the one parameter family of Fano threefolds $V_{12}^m$ of Picard rank $1$ and genus $12$ that admit $\mathbb C^*$ action. In particular we improve the bound on the log canonical thresholds for such manifolds. We show that any threefold from $V_{12}^m$ admits an additional symmetry which anti-commutes with the $\mathbb C^*$ action, a fact that was previously observed near the Mukai–Umemura threefold by Rollin, Simanca, and Tipler. As a consequence the Kähler–Einstein manifolds in the class form an open subset in the standard topology. Moreover, we find an explicit description for all Fano threefolds of genus $12$ and Picard number $1$ in terms of the quartic associated to the variety-of-sum-of-powers construction. We describe explicitly the Hilbert scheme of lines on such Fano threefolds.

Key words and phrases: Fano threefold, log canonical threshold, Kähler–Einstein metric.

Funding Agency Grant Number
NCN 2013/08/A/ST1/00312
Iuventus plus 0301/IP3/2015/73
The first and second named authors were supported by NCN grant 2013/08/A/ST1/00312; the third named author was supported by the grant Iuventus plus 0301/IP3/2015/73 “Teoria reprezentacji oraz wlasności rozmaitości siecznych”.


DOI: https://doi.org/10.17323/1609-4514-2017-17-1-15-33

Full text: http://www.mathjournals.org/.../2017-017-001-002.html
References: PDF file   HTML file

Bibliographic databases:

MSC: Primary 32Q20; Secondary 32U15, 32G05
Received: October 14, 2015; in revised form September 8, 2016
Language:

Citation: Sławomir Dinew, Grzegorz Kapustka, Michał Kapustka, “Remarks on Mukai threefolds admitting $\mathbb C^*$ action”, Mosc. Math. J., 17:1 (2017), 15–33

Citation in format AMSBIB
\Bibitem{DinKapKap17}
\by S{\l}awomir~Dinew, Grzegorz~Kapustka, Micha{\l}~Kapustka
\paper Remarks on Mukai threefolds admitting $\mathbb C^*$ action
\jour Mosc. Math.~J.
\yr 2017
\vol 17
\issue 1
\pages 15--33
\mathnet{http://mi.mathnet.ru/mmj623}
\crossref{https://doi.org/10.17323/1609-4514-2017-17-1-15-33}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3634518}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000402641900002}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. Kuznetsov, Yu. Prokhorov, “Prime Fano threefolds of genus $12$ with a $\mathbb{G}_{\mathrm{m}}$-action and their automorphisms”, Epijournal Geom. Algebr., 2 (2018), 3  mathscinet  isi
    2. A. G. Kuznetsov, Yu. G. Prokhorov, C. A. Shramov, “Hilbert schemes of lines and conics and automorphism groups of Fano threefolds”, Jap. J. Math., 13:1 (2018), 109–185  crossref  mathscinet  zmath  isi  scopus
  • Moscow Mathematical Journal
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