Fanography

4-13 5-2

Fano variety 5-1

blowup of 2-29 in the disjoint union of three exceptional lines of the blowup

Picard rank

5 (others)

$-\mathrm{K}_X^3$

28

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

0

index

1

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

17

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

yes

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

yes

hyperelliptic

no

trigonal

no

This variety is rational.

This variety is the blowup of

• 4-4, in a curve of genus 0
• 4-12, 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

#100

Fanosearch

#114

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

$\mathbb{P}^1 \times \mathbb{P}^3 \times \mathbb{P}^{8} \times \mathbb{P}^{11}$

bundle

$\mathcal{O}(1,1,0,0) \oplus \Lambda(0,0,1,0) \oplus \mathcal{O}(-1,1,1,0) \oplus \mathcal{Q}_{\mathbb{P}^3}(0,0,0,1) \oplus \mathcal{Q}_{\mathbb{P}^8}(0,0,0,1)$

See the big table for more information.

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

• the book