Fanography

2-29 2-31

Fano variety 2-30

blowup of 1-17 in a conic

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

• 1-17, in a curve of genus 0

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

• 3-18, in a curve of genus 0
• 3-23, in a curve of genus 0

number of moduli

0

Bott vanishing

holds

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

GRDB

#4

Fanosearch

#70

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

$\mathbb{P}^3 \times \mathbb{P}^4$

bundle

$\mathcal{Q}_{\mathbb{P}^3}(0,1) \oplus \mathcal{O}(1,1)$

variety

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

bundle

$\mathcal{Q}_2 \oplus \mathcal{O}(1,1)$

See the big table for more information.

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

• the book