Identification
Hodge diamond and polyvector parallelogram
1
0 0
0 1 0
0 21 21 0
0 1 0
0 0
1
0 0
0 1 0
0 21 21 0
0 1 0
0 0
1
1
0 0
0 34 3
0 0 0 7
0 7 0
0 0
0
0 0
0 34 3
0 0 0 7
0 7 0
0 0
0
Anticanonical bundle
- index
- 2
- del Pezzo of degree 1
- $X\to\mathbb{P}^2$
- $\dim\mathrm{H}^0(X,\omega_X^\vee)$
- 7
- $-\mathrm{K}_X$ very ample?
- no
- $-\mathrm{K}_X$ basepoint free?
- yes
- hyperelliptic
- no
- trigonal
- no
Birational geometry
This variety is not known to be unirational.
This variety is primitive.
This variety can be blown up (in a curve) to
- 2-1, in a curve of genus 1
This variety is fibre-like, i.e. it can appear as the fibre of a Mori fibre space.
Deformation theory
- number of moduli
- 34
- Bott vanishing
- does not hold
$\mathrm{Aut}^0(X)$ | $\dim\mathrm{Aut}^0(X)$ | number of moduli |
---|---|---|
$0$ | 0 | 34 |
Period sequence
There is no period sequence associated to this Fano 3-fold.
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
- $\mathbb{P}(1^3,2,3)$
- bundle
- $\mathcal{O}(6)$
See the big table for more information.
K-stability
- every member is K‑stable
- every member is K‑polystable
- every member is K‑semistable