Group actions on the Fano-Mukai $4$-fold

Question

Do any of the Fano-Mukai $4$-fold from https://arxiv.org/pdf/1706.04926.pdf have a $\mathbb{C}^{*}$-action with isolated fixed points? If so is it possible to compute the weights of the action?

Answer 1

One of these fourfolds has a $GL_2$-action. The corresponding group $GL_2$ is a subgroup in the group $G_2$ of automorphisms of the genus-10 fivefold. In particular, they have the same maximal torus $T$.

Now, the $T$-action on the genus-10 fivefold has isolated fixed points (this is true for the maximal torus action on any homogeneous space of a semisimple algebraic group). Moreover, these points can be explicitly described together with the $T$-action on the tangent spaces. Next, the fourfold is a hyperplane section of the fivefold, hence each $T$-fixed point of the formed is also fixed on the latter, and since their numbers agree, they just coincide. This allows to describe the points and compute the $T$-action on tangent spaces.

Finally, for a general $G_m$ inside $T$ the $G_m$-fixed points are the same as $T$-fixed points, and the $G_m$-action on tangent spaces is obtained by restriction.