Fanography

1-14 1-16

Fano variety 1-15

section of Plücker embedding of $\mathrm{Gr}(2,5)$ by codimension 3 subspace

Picard rank

1 (others)

$-\mathrm{K}_X^3$

40

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

0

index

2

del Pezzo of degree 5

$X\hookrightarrow\mathbb{P}^{6}$

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

23

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

yes

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

yes

hyperelliptic

no

trigonal

no

This variety is rational.

This variety is primitive.

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

• 2-14, in a curve of genus 1
• 2-20, in a curve of genus 0
• 2-22, in a curve of genus 0
• 2-26, 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

0

Bott vanishing

does not hold

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

GRDB

#46

Fanosearch

#14

A full exceptional collection was constructed by Orlov in 1991, see [MR1294662] .

Generic semisimplicity of:

• small quantum cohomology, proved by someone in at some point, see [?] , using

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

variety

$\operatorname{Gr}(2,5)$

bundle

$\mathcal{O}(1)^{\oplus 3}$

See the big table for more information.

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

• the book

The Hilbert scheme of lines is $\mathbb{P}^2$.

Its Hodge diamond is