%0 Journal Article
%T Remarks on Mukai threefolds admitting $C^{*}$ action
%A Slawomir Dinew
%A Grzegorz Kapustka
%A Michal Kapustka
%J Mathematics
%D 2015
%I arXiv
%X We investigate geometric invariants of the one parameter family of Mukai threefolds that admit $\mathbb C^{*}$ action. In particular we find the invariant divisors in the anticanonical system, and thus establish a bound on the log canonical thresholds. Furthermore we find an explicit description of such threefolds in terms of the quartic associated to the variety-of-sum-of-powers construction. This yields that any such threefold admits an additional symmetry which anticommutes 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 K\"ahler-Einstein manifolds in the class form an open subset in the standard topology.
%U http://arxiv.org/abs/1506.03286v1