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

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.