Moscow Mathematical Journal

Volume 17, Issue 1, January–March 2017 pp. 15–33.

Remarks on Mukai Threefolds Admitting ℂ^* Action

Authors: Sławomir Dinew (1), Grzegorz Kapustka (2) and Michał Kapustka (3)
Author institution:(1) Department of Mathematics and Computer Science, Jagiellonian University, Kraków, Poland
(2) Institute of Mathematics of the Polish Academy of Sciences, Warsaw
Department of Mathematics and Computer Science, Jagiellonian University, Kraków, Poland
(3) University of Stavanger, Norway

Summary:

We investigate geometric properties of the one parameter family of Fano threefolds V₁₂^m of Picard rank 1 and genus 12 that admit ℂ^* action. In particular we improve the bound on the log canonical thresholds for such manifolds. We show that any threefold from V₁₂^m admits an additional symmetry which anti-commutes with the 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.

2010 Math. Subj. Class. Primary: 32Q20; Secondary: 32U15, 32G05.

Keywords: Fano threefold, log canonical threshold, Kähler–Einstein metric.

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