0 0
0 1 0
0 10 10 0
0 1 0
0 0
1
0 0
0 22 0
0 0 0 8
0 1 0
0 0
0
No nontrivial holomorphic Poisson structures.
See Loray–Pereira–Touzet.
- 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
This variety is generally not rational.
A very general member is not stably rational.
The Weyl group associated to the KKMR decomposition is $\{e\}$; see the Weyl-group table for sources and conventions.
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
- 22
- 19
- Bott vanishing
- does not hold
| $\mathrm{Aut}^0(X)$ | $\dim\mathrm{Aut}^0(X)$ | number of moduli |
|---|---|---|
| $0$ | 0 | 22 |
The naive atomic decomposition has atom dimensions $2+1+1$.
A standard semiorthogonal decomposition is $\langle \operatorname{Ku}(X), \mathcal{E}_2, \mathcal{O}_X \rangle$.
Both decompositions are recorded in Böhning–Graf von Bothmer–Su'a.
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
- $\operatorname{Gr}(2,5)$
- bundle
- $\mathcal{O}(2) \oplus \mathcal{O}(1)^{\oplus2}$
See the big table for more information.
- every member is K‑stable
- every member is K‑polystable
- every member is K‑semistable
For this deformation family, the general member has coregularity at most 1; some member has coregularity 0.
The generic and uniform statements are due to Avilov–Loginov–Przyjalkowski; the existence of a smooth member of coregularity zero in every family is due to Zhakupov.
See coregularity for terminology.
For every K-polystable member, the Futaki invariant vanishes identically on the entire Kähler cone.
This is the computation of Sektnan–Tipler.