Identification
Fano variety 28
 double cover of 235 with branch locus an anticanonical divisor such that the intersection with the exceptional divisor is smooth
 double cover of 235 with branch locus an anticanonical divisor such that the intersection with the exceptional divisor is singular but reduced
 Picard rank
 2 (others)
 $\mathrm{K}_X^3$
 14
 $\mathrm{h}^{1,2}(X)$
 9
Hodge diamond and polyvector parallelogram
1
0 0
0 2 0
0 9 9 0
0 2 0
0 0
1
0 0
0 2 0
0 9 9 0
0 2 0
0 0
1
1
0 0
0 18 3
0 0 1 10
0 1 0
0 0
0
0 0
0 18 3
0 0 1 10
0 1 0
0 0
0
Anticanonical bundle
 index
 1
 $\dim\mathrm{H}^0(X,\omega_X^\vee)$
 10
 $\mathrm{K}_X$ very ample?
 yes
 $\mathrm{K}_X$ basepoint free?
 yes
 hyperelliptic
 no
 trigonal
 no
Birational geometry
This variety is not rational but unirational.
This variety is primitive.
Deformation theory
 number of moduli

 18
 17
 Bott vanishing
 does not hold
$\mathrm{Aut}^0(X)$  $\dim\mathrm{Aut}^0(X)$  number of moduli 

$0$  0  18 
Period sequence
Extremal contractions
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
 $\mathbb{P}^2 \times \mathbb{P}^3 \times \mathbb{P}^{12}$
 bundle
 $\Lambda(0,0,1) \oplus \mathcal{O}(0,0,2)$
See the big table for more information.
Kstability
 Kstability is not understood
 every member is Kpolystable
 every member is Ksemistable
See the big table for more information.