Fanography

2-2 2-4

Fano variety 2-3

blowup of 1-12 in an elliptic curve which is the intersection of two divisors from half the anticanonical linear system

Picard rank

2 (others)

$-\mathrm{K}_X^3$

8

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

11

index

1

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

7

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

no

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

yes

hyperelliptic

yes

trigonal

no

This variety is not rational but unirational.

This variety is the blowup of

• 1-12, in a curve of genus 1

number of moduli

23

Bott vanishing

does not hold

There is no period sequence associated to this Fano 3-fold.

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

$\mathbb{P}(1^4,2) \times \mathbb{P}^1$

bundle

$\mathcal{O}(4,0) \oplus \mathcal{O}(1,1)$

variety

$\mathbb{P}^3 \times \mathbb{P}^{10} \times \mathbb{P}^1$

bundle

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

See the big table for more information.