Fanography

9-1

Fano variety 10-1

$\mathbb{P}^1\times\mathrm{Bl}_8\mathbb{P}^2$

Picard rank

10

$-\mathrm{K}_X^3$

6

$\mathrm{h}^{1,2}(X)$

0

Mori–Mukai $d$

-256

Still open.

index

1

$\dim\mathrm{H}^0(X,\omega_X^\vee)$

6

$-\mathrm{K}_X$ very ample?

no

$-\mathrm{K}_X$ basepoint free?

no

hyperelliptic

yes

trigonal

no

This variety is rational.

This variety is stably rational.

The Weyl group associated to the KKMR decomposition is $\mathrm{E}_8$; see the Weyl-group table for sources and conventions.

This variety is the blowup of

• 9-1, in a curve of genus 0

number of moduli

8

Bott vanishing

does not hold

There is no period sequence associated to this Fano 3-fold.

$\mathbb{P}^1$-bundle over $\mathrm{Bl}_8\mathbb{P}^2$, for the vector bundle $\mathcal{O}_{\mathrm{Bl}_8\mathbb{P}^2}\oplus\mathcal{O}_{\mathrm{Bl}_8\mathbb{P}^2}$.

A full exceptional collection can be constructed using Orlov's blowup formula.

Generic semisimplicity of:

• small quantum cohomology, proved by Ciolli in 2005, see [MR2168069] , using the description of quantum cohomology of a $\mathbb{P}^1$-bundle

Fano 3-folds from homogeneous vector bundles over Grassmannians gives the following description(s):

variety

$\mathbb{P}(1^2,2,3) \times \mathbb{P}^1$

bundle

$\mathcal{O}(6,0)$

See the big table for more information.

• none are K-stable
• every member is K‑polystable
• every member is K‑semistable

• the book

For this deformation family, the general member has coregularity 0.

The generic and uniform statements are due to Avilov–Loginov–Przyjalkowski; the existence of a smooth member of coregularity zero in every family is due to Zhakupov.

See coregularity for terminology.

For every K-polystable member, the Futaki invariant vanishes identically on the entire Kähler cone.

This is the computation of Sektnan–Tipler.