Fanography

2-15 2-17

Fano variety 2-16

blowup of 1-14 in a conic

Picard rank

2 (others)

$-\mathrm{K}_X^3$

22

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

2

index

1

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

14

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

yes

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

yes

hyperelliptic

no

trigonal

no

This variety is rational.

This variety is the blowup of

• 1-14, in a curve of genus 0

number of moduli

7

Bott vanishing

does not hold

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

GRDB

#104

Fanosearch

#59

There exist interesting semiorthogonal decompositions, but this data is not yet added.

By Hertling–Manin–Teleman we have that quantum cohomology cannot be generically semisimple, as $\mathrm{h}^{1,2}\neq 0$.

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

variety

$\mathbb{P}^2 \times \operatorname{Gr}(2,4)$

bundle

$\mathcal{U}^{\vee}_{\operatorname{Gr}(2,4)}(1,0) \oplus \mathcal{O}(0,2)$

variety

$\operatorname{Fl}(1,2,4)$

bundle

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

See the big table for more information.

• general member is K‑stable
• general member is K‑polystable
• general member is K‑semistable

• the book