Fanography

3-1 3-3

Fano variety 3-2

divisor from $|\mathcal{L}^{\otimes 2}\otimes\mathcal{O}(2,3)|$ on the $\mathbb{P}^2$-bundle $\mathbb{P}(\mathcal{O}\oplus\mathcal{O}(-1,-1)^{\oplus 2})$ over $\mathbb{P}^1\times\mathbb{P}^1$ such that $X\cap Y$ is irreducible, and $\mathcal{L}$ is the tautological bundle, and $Y\in|\mathcal{L}|$

Picard rank

3 (others)

$-\mathrm{K}_X^3$

14

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

3

index

1

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

10

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

yes

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

yes

hyperelliptic

no

trigonal

yes

This variety is rational.

This variety is primitive.

number of moduli

11

Bott vanishing

does not hold

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

GRDB

#157

Fanosearch

#97

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

$\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^5$

bundle

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

See the big table for more information.

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

• the book
• 2403.03700