0 0
0 1 0
0 3 3 0
0 1 0
0 0
1
0 0
0 12 0
0 0 0 11
0 0 0
0 0
0
- index
- 1
- $\dim\mathrm{H}^0(X,\omega_X^\vee)$
- 11
- $-\mathrm{K}_X$ 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
- 12
- Bott vanishing
- does not hold
$\mathrm{Aut}^0(X)$ | $\dim\mathrm{Aut}^0(X)$ | number of moduli |
---|---|---|
$0$ | 0 | 12 |
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
- $\operatorname{Gr}(3,6)$
- bundle
- $\bigwedge^2\mathcal{U}^{\vee} \oplus \mathcal{O}(1)^{\oplus3}$
See the big table for more information.
- every member is K‑stable
- every member is K‑polystable
- every member is K‑semistable
See the big table for more information.
The Hilbert scheme of conics is the ruled surface obtained from projectivisation of simple rank 2 bundle over a smooth curve of genus 3.
Its Hodge diamond is
3 3
0 2 0
3 3
1