Fanography

1-5 1-7

Fano variety 1-6

section of half-spinor embedding of a connected component of $\mathrm{OGr}_+(5,10)$ by codimension 7 subspace

Picard rank

1 (others)

$-\mathrm{K}_X^3$

12

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

7

index

1

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

9

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

yes

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

yes

hyperelliptic

no

trigonal

no

This variety is rational.

This variety is primitive.

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

number of moduli

18

Bott vanishing

does not hold

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

GRDB

#150

Fanosearch

#7

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

$\operatorname{OGr}^+(5,10)$

bundle

$\mathcal{O}(\frac{1}{2})^{\oplus7}$

variety

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

bundle

$\mathcal{U}^{\vee}(1)\oplus \mathcal{O}(1)$

See the big table for more information.

The Hilbert scheme of conics is the symmetric square of a smooth curve of genus 7.

Its Hodge diamond is