Fanography

4-7 4-9

Fano variety 4-8

blowup of 3-27 in a curve of degree $(0,1,1)$

Picard rank

4 (others)

$-\mathrm{K}_X^3$

38

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

0

index

1

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

22

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

yes

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

yes

hyperelliptic

no

trigonal

no

This variety is rational.

This variety is the blowup of

• 3-27, in a curve of genus 0
• 3-31, 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

#54

Fanosearch

#105

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

$\mathbb{P}^1 \times \mathbb{P}^3 \times \mathbb{P}^4$

bundle

$\mathcal{O}(1,0,1) \oplus \mathcal{O}(0,2,0) \oplus \mathcal{Q}_{\mathbb{P}^3}(0,0,1)$

variety

$\mathbb{P}^1 \times \operatorname{Fl}(1,2,5)$

bundle

$\mathcal{O}(1;1,0) \oplus \mathcal{O}(0;0,2) \oplus \mathcal{Q}_2$

See the big table for more information.

• none are K-stable
• none are K-polystable
• none are K-semistable

• the book