Flexible affine cones and flexible coverings

@article{Michaek2018FlexibleAC,
  title={Flexible affine cones and flexible coverings},
  author={Mateusz Michałek and Alexander Perepechko and Hendrik S{\"u}{\ss}},
  journal={Mathematische Zeitschrift},
  year={2018},
  volume={290},
  pages={1457-1478}
}
We provide a new criterion for flexibility of affine cones over varieties covered by flexible affine varieties. We apply this criterion to prove flexibility of affine cones over secant varieties of Segre–Veronese embeddings and over certain Fano threefolds. We further prove flexibility of total coordinate spaces of Cox rings of del Pezzo surfaces. 

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References: [1] H. Ahmadinezhad, I. Cheltsov, and J. Schicho, On a conjecture of Tian.Math. Z.288(2018), no. 1-2, 217-241 Zbl1390.14109MR3774411 · Zbl 1390.14109 [2] J. Alper, H. Blum, D.

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References

SHOWING 1-10 OF 54 REFERENCES

Flexible affine cones over del Pezzo surfaces of degree 4

For an arbitrary ample divisor A on a smooth del Pezzo surface S of degree 4, we show that the affine cone of S defined by A is flexible.

Flexibility of affine cones over del Pezzo surfaces of degree 4 and 5

We prove that the action of the special automorphism group on affine cones over del Pezzo surfaces of degree 4 and 5 is infinitely transitive.

Group actions on affine cones

We address the following question: Determine the affine cones over smooth projective varieties which admit an action of a connected algebraic group different from the standard C*-action by scalar

Affine cones over smooth cubic surfaces

We show that affine cones over smooth cubic surfaces do not admit non-trivial $\mathbb{G}_a$ -actions.

Deformations of rational T-varieties

We show how to construct certain homogeneous deformations for rational normal varieties with codimension one torus action. This can then be used to construct homogeneous deformations of any toric

Tangents and Secants of Algebraic Varieties

Theorem on tangencies and Gauss maps Projections of algebraic varieties Varieties of small codimension corresponding to orbits of algebraic groups Severi varieties Linear systems of hyperplane

Infinite transitivity on universal torsors

TLDR
It is proved that the automorphism group of the quasiaffine variety X' acts on X' infinitely transitively and wide classes of varieties X admitting such a covering are found.

The Geometry of T-Varieties

This is a survey of the language of polyhedral divisors describing T-varieties. This language is explained in parallel to the well established the- ory of toric varieties. In addition to basic

Gluing Affine Torus Actions Via Divisorial Fans

Generalizing the passage from a fan to a toric variety, we provide a combinatorial approach to construct arbitrary effective torus actions on normal, algebraic varieties. Based on the notion of a
...