Identification
Hodge diamond and polyvector parallelogram
1
0 0
0 1 0
0 10 10 0
0 1 0
0 0
1
0 0
0 1 0
0 10 10 0
0 1 0
0 0
1
1
0 0
0 22 0
0 0 0 8
0 1 0
0 0
0
0 0
0 22 0
0 0 0 8
0 1 0
0 0
0
Anticanonical bundle
 index
 1
 $\dim\mathrm{H}^0(X,\omega_X^\vee)$
 8
 $\mathrm{K}_X$ very ample?
 yes if (a), else no
 $\mathrm{K}_X$ basepoint free?
 yes
 hyperelliptic
 no
 trigonal
 no
Birational geometry
This variety is known to be generically nonrational.
This variety is primitive.
This variety is fibrelike, i.e. it can appear as the fibre of a Mori fibre space.
Deformation theory
 number of moduli

 22
 19
 Bott vanishing
 does not hold
$\mathrm{Aut}^0(X)$  $\dim\mathrm{Aut}^0(X)$  number of moduli 

$0$  0  22 
Period sequence
Semiorthogonal decompositions
There exist interesting semiorthogonal decompositions, but this data is not yet added.
Structure of quantum cohomology
By Hertling–Manin–Teleman we have that quantum cohomology cannot be generically semisimple, as $\mathrm{h}^{1,2}\neq 0$.
Zero section description
Fano 3folds from homogeneous vector bundles over Grassmannians gives the following description(s):
 variety
 $\operatorname{Gr}(2,5)$
 bundle
 $\mathcal{O}(2) \oplus \mathcal{O}(1)^{\oplus2}$
See the big table for more information.
Kstability
 every member is K‑stable
 every member is K‑polystable
 every member is K‑semistable
See the big table for more information.