Fanography

2-30 2-32

Fano variety 2-31

blowup of 1-16 in a line

Picard rank

2 (others)

$-\mathrm{K}_X^3$

46

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

0

index

1

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

26

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

yes

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

yes

hyperelliptic

no

trigonal

no

This variety is rational.

This variety is the blowup of

• 1-16, in a curve of genus 0

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

• 3-15, in a curve of genus 0
• 3-20, in a curve of genus 0
• 3-23, in a curve of genus 0

number of moduli

0

Bott vanishing

does not hold

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

GRDB

#15

Fanosearch

#48

$\mathbb{P}^1$-bundle over $\mathbb{P}^2$, for the vector bundle $0\to\mathcal{O}_{\mathbb{P}^2}\to\mathcal{E}\to\mathcal{J}_x\to0$.

A full exceptional collection can be constructed using Orlov's blowup formula.

Generic semisimplicity of:

• small quantum cohomology, proved by Ciolli in 2005, see [MR2168069] , using the description of quantum cohomology of a $\mathbb{P}^1$-bundle

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

variety

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

bundle

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

See the big table for more information.

• none are K-stable
• none are K-polystable
• none are K-semistable

• the book