Fanography

1-13 1-15

Fano variety 1-14

complete intersection of 2 quadrics in $\mathbb{P}^5$

Picard rank

1 (others)

$-\mathrm{K}_X^3$

32

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

2

index

2

del Pezzo of degree 4

$X\hookrightarrow\mathbb{P}^{5}$

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

19

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

yes

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

yes

hyperelliptic

no

trigonal

no

This variety is rational.

This variety is primitive.

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

• 2-10, in a curve of genus 1
• 2-16, in a curve of genus 0
• 2-19, in a curve of genus 0

This variety is fibre-like, i.e. it can appear as the fibre of a Mori fibre space.

number of moduli

3

Bott vanishing

does not hold

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

GRDB

#75

Fanosearch

#2

There exist interesting semiorthogonal decompositions, but this data is not yet added.

By Hertling–Manin–Teleman we have that quantum cohomology cannot be generically semisimple, as $\mathrm{h}^{1,2}\neq 0$.

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

variety

$\mathbb{P}^5$

bundle

$\mathcal{O}(2)^{\oplus 2}$

See the big table for more information.

• every member is K‑stable
• every member is K‑polystable
• every member is K‑semistable

• the book

The Hilbert scheme of lines is an abelian surface.

Its Hodge diamond is