Fanography

2-21 2-23

Fano variety 2-22

blowup of 1-15 in a conic

Picard rank

2 (others)

$-\mathrm{K}_X^3$

30

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

0

index

1

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

18

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

yes

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

yes

hyperelliptic

no

trigonal

no

This variety is rational.

This variety is the blowup of

• 1-15, in a curve of genus 0
• 1-17, in a curve of genus 0

number of moduli

1

Bott vanishing

does not hold

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

GRDB

#69

Fanosearch

#50

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 a 1 or 2-curve blowup of $\mathbb{P}^3$ or $Q^3$

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

variety

$\mathbb{P}^3 \times \operatorname{Gr}(2,5)$

bundle

$\mathcal{Q}_{\operatorname{Gr}(2,5)}(1,0) \oplus \mathcal{O}(0,1)^{\oplus 3}$

variety

$\operatorname{Fl}(1,2,5)$

bundle

$\mathcal{O}(1,0) \oplus \mathcal{O}(0,1)^{\oplus 3}$

See the big table for more information.

• general member is K‑stable but there exists one that is not
• general member is K‑polystable but there exists one that is not
• every member is K‑semistable

• the book
• 2110.14762