Fanography

3-19 3-21

Fano variety 3-20

blowup of 1-16 in the disjoint union of two lines

Picard rank

3 (others)

$-\mathrm{K}_X^3$

38

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

0

Mori–Mukai $d$

28

Still open.

index

1

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

22

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

yes

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

yes

hyperelliptic

no

trigonal

no

This variety is rational.

This variety is stably rational.

The Weyl group associated to the KKMR decomposition is $\mathrm{A}_1$; see the Weyl-group table for sources and conventions.

This variety is the blowup of

• 2-31, in a curve of genus 0
• 2-32, 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

#38

Fanosearch

#63

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

$\mathbb{P}^2_1 \times \mathbb{P}^2_2 \times \mathbb{P}^4$

bundle

$\mathcal{O}(1,1,0) \oplus \mathcal{Q}_{\mathbb{P}^2_1}(0,0,1) \oplus \mathcal{Q}_{\mathbb{P}^2_2}(0,0,1)$

See the big table for more information.

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

• the book

For this deformation family, every smooth member has coregularity 0.

The generic and uniform statements are due to Avilov–Loginov–Przyjalkowski; the existence of a smooth member of coregularity zero in every family is due to Zhakupov.

See coregularity for terminology.

For every K-polystable member, the zero locus of the Futaki invariant contains a 2-dimensional family of Kähler classes including the anticanonical class.

This is the computation of Sektnan–Tipler.