V. Batyrev: “Toric Deformations of Some Spherical Fano Varieties”

Here are my (incomplete) notes from Batyrev’s talk in the Extremal Laurent Polynomials workshop at Imperial:


If you have complete notes, please post them here.

Partial summary:

  • spherical varieties
  • a model family of examples: SL_2-varieties
  • review: compactification and toric varieties
  • spherical compactification in these examples
  • valuations and coloured cones
  • finding toric degenerations

One Comment

  1. Sergey says:

    Maybe it worth to note.

    Yuri Prokhorov explianed to me that 2.21 with involution is actually a spherical variety – it has finite number of orbits with respect to G = PGL(2,\mathbb{C}).
    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 \PP^1 is embedded by linear system O(4) into \PP^4 = \PP(|O(4)|^*),
    and its automorphism group acts on \PP^4 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?

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