Fanography

4-6 4-8

Fano variety 4-7

blowup of 2-32 in the disjoint union of a curve of degree $(0,1)$ and a curve of degree $(1,0)$

Picard rank

4 (others)

$-\mathrm{K}_X^3$

36

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

0

index

1

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

21

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

yes

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

yes

hyperelliptic

no

trigonal

no

This variety is rational.

This variety is the blowup of

• 3-24, in a curve of genus 0
• 3-28, in a curve of genus 0

number of moduli

0

Bott vanishing

holds

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

GRDB

#57

Fanosearch

#69

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

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

bundle

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

variety

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

bundle

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

See the big table for more information.

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

• the book