Fanography

3-7 3-9

Fano variety 3-8

divisor from the linear system $|(\alpha\circ\pi_1)^*(\mathcal{O}_{\mathbb{P}^2}(1)\otimes\pi_2^*(\mathcal{O}_{\mathbb{P}^2}(2))|$ where $\pi_i\colon\mathrm{Bl}_1\mathbb{P}^2\times\mathbb{P}^2\to\mathrm{Bl}_1\mathbb{P}^2,\mathbb{P}^2$ are the projections, and $\alpha\colon\mathrm{Bl}_1\mathbb{P}^2\to\mathbb{P}^2$ is the blowup

Picard rank

3 (others)

$-\mathrm{K}_X^3$

24

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

0

Mori–Mukai $d$

34

Still open.

index

1

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

15

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

yes

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

yes

hyperelliptic

no

trigonal

no

This variety is rational.

This variety is stably rational.

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

This variety is the blowup of

• 2-24, in a curve of genus 0
• 2-34, in a curve of genus 0

number of moduli

3

Bott vanishing

does not hold

The following period sequences are associated to this Fano 3-fold:

GRDB

#112

Fanosearch

#85

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

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

variety

$\mathbb{P}^1 \times \mathbb{P}^2 \times \mathbb{P}^2$

bundle

$\mathcal{O}(0,1,2) \oplus \mathcal{O}(1,1,0)$

See the big table for more information.

• general member is K‑stable but there exist members that are not
• general member is K‑polystable but there exist members that are not
• general member is K‑semistable

• the book

For this deformation family, every smooth 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.