Fanography

3-17 3-19

Fano variety 3-18

blowup of 1-17 in the disjoint union of a line and a conic

Picard rank

3 (others)

$-\mathrm{K}_X^3$

36

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

0

index

1

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

21

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

yes

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

yes

hyperelliptic

no

trigonal

no

This variety is rational.

This variety is the blowup of

• 2-29, in a curve of genus 0
• 2-30, in a curve of genus 0
• 2-33, in a curve of genus 0

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

• 4-4, 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

#41

Fanosearch

#160

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

$\mathbb{P}^1\times \mathbb{P}^3 \times \mathbb{P}^4$

bundle

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

variety

$\mathbb{P}^1 \times \operatorname{Fl}(1,2,5)$

bundle

$\mathcal{Q}_2(0;0,0) \oplus \mathcal{O}(0;1,1) \oplus \mathcal{O}(1;0,1)$

See the big table for more information.

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

• the book