Fanography

3-8 3-10

Fano variety 3-9

blowup of the cone over the Veronese of $\mathbb{P}^2$ in $\mathbb{P}^5$ with center the disjoint union of the vertex and a quartic curve on $\mathbb{P}^2$

Picard rank

3 (others)

$-\mathrm{K}_X^3$

26

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

3

index

1

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

16

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

yes

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

yes

hyperelliptic

no

trigonal

no

This variety is rational.

This variety is the blowup of

• 2-36, in a curve of genus 3

number of moduli

6

Bott vanishing

does not hold

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

GRDB

#22

Fanosearch

#68

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

$\mathbb{P}^2 \times \mathbb{P}^6 \times \mathbb{P}^{20}$

bundle

$\Lambda(0,1,0) \oplus \mathcal{Q}_{\mathbb{P}^6}(0,0,1) \oplus K(0,0,1)$

See the big table for more information.

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

• the book