Yuri Prokhorov explianed to me that 2.21 with involution is actually a spherical variety – it has finite number of orbits with respect to .

Construction is the following (http://books.google.com/books?id=iZOJuKZTEJkC&lpg=PA137&ots=OTpkhewIeD&dq=Tetsuo%20Nakano%20%22equivariant%20completions%22&lr&hl=ru&pg=PA143#v=onepage&q=Tetsuo%20Nakano%20%22equivariant%20completions%22&f=false):

Projective line is embedded by linear system into ,

and its automorphism group acts on preserving its image (twisted quartic C).

In fact it is just 4th symmetric power of tautological representation.

Choose a generic invariant of degree $2$ i.e. smooth three-dimensional quadric Q that contains C and is G-invariant.

Then blowup of Q in C is a Fano threefold V of type 2.21 with action of G, the other contraction is also G-invariant and hence isomorphic to the original one,

so automorphisms of V are extended from G by an involution.

Numerous toric degenerations of spherical varieties were constructed by Alexeev and Brion,

however I don’t know if there is a theorem describing quantum multiplication by $c_1$ for spherical varieties.

Morally it should be possible to do using same technique as for toric and homogeneous varieties – just localize curves to B-invariant,

but have anybody already wrote on this? Could it be in the missing part of the talk?