Fanography

3-18 3-20

Fano variety 3-19

blowup of 1-16 in two non-collinear points

Picard rank

3 (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

• 2-35, in a curve of genus 0

This variety can be blown up (in a curve) to

• 4-4, 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

#18

Fanosearch

#57

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

$\mathbb{P}^2 \times \mathbb{P}^4$

bundle

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

See the big table for more information.

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

• the book