Identification
Hodge diamond and polyvector parallelogram
1
0 0
0 2 0
0 11 11 0
0 2 0
0 0
1
0 0
0 2 0
0 11 11 0
0 2 0
0 0
1
1
0 0
0 23 1
0 0 3 7
0 1 0
0 0
0
0 0
0 23 1
0 0 3 7
0 1 0
0 0
0
Anticanonical bundle
- index
- 1
- $\dim\mathrm{H}^0(X,\omega_X^\vee)$
- 7
- $-\mathrm{K}_X$ very ample?
- no
- $-\mathrm{K}_X$ basepoint free?
- yes
- hyperelliptic
- yes
- trigonal
- no
Birational geometry
This variety is not rational but unirational.
This variety is the blowup of
- 1-12, in a curve of genus 1
Deformation theory
- number of moduli
- 23
- Bott vanishing
- does not hold
$\mathrm{Aut}^0(X)$ | $\dim\mathrm{Aut}^0(X)$ | number of moduli |
---|---|---|
$0$ | 0 | 23 |
Period sequence
There is no period sequence associated to this Fano 3-fold.
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 3-folds from homogeneous vector bundles over Grassmannians gives the following description(s):
- variety
- $\mathbb{P}(1^4,2) \times \mathbb{P}^1$
- bundle
- $\mathcal{O}(4,0) \oplus \mathcal{O}(1,1)$
- variety
- $\mathbb{P}^3 \times \mathbb{P}^{10} \times \mathbb{P}^1$
- bundle
- $\Lambda(0,1,0) \oplus \mathcal{O}(0,2,0) \oplus \mathcal{O}(1,0,1)$
See the big table for more information.
K-stability
- every member is K‑stable
- every member is K‑polystable
- every member is K‑semistable