# 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
*V*_{12}^{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 *V*_{12}^{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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