Fanography

2-28 2-30

Fano variety 2-29

blowup of 1-16 in a conic

Picard rank

2 (others)

$-\mathrm{K}_X^3$

40

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

0

index

1

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

23

$-\mathrm{K}_X$ 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

• 3-10, in a curve of genus 0
• 3-15, in a curve of genus 0
• 3-18, 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

#35

Fanosearch

#42

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

$\mathbb{P}^1 \times \mathbb{P}^4$

bundle

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

See the big table for more information.

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

• the book