0 0
0 4 0
0 1 1 0
0 4 0
0 0
1
0 1
0 2 7
0 0 0 17
0 0 0
0 0
0
Still open.
- index
- 1
- $\dim\mathrm{H}^0(X,\omega_X^\vee)$
- 17
- $-\mathrm{K}_X$ very ample?
- yes
- $-\mathrm{K}_X$ basepoint free?
- yes
- hyperelliptic
- no
- trigonal
- no
This variety is rational.
This variety is stably rational.
The Weyl group associated to the KKMR decomposition is $\mathrm{A}_1\times\mathrm{A}_1$; see the Weyl-group table for sources and conventions.
This variety is the blowup of
- 3-31, in a curve of genus 1
- number of moduli
- 2
- Bott vanishing
- does not hold
| $\mathrm{Aut}^0(X)$ | $\dim\mathrm{Aut}^0(X)$ | number of moduli |
|---|---|---|
| $\mathbb{G}_{\mathrm{m}}$ | 1 | 2 |
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
- $\mathbb{P}^3 \times \mathbb{P}^4 \times \mathbb{P}^5$
- bundle
- $\mathcal{Q}_{\mathbb{P}^3}(0,1,0) \oplus \mathcal{Q}_{\mathbb{P}^4}(0,0,1) \oplus \mathcal{O}(2,0,0) \oplus \mathcal{O}(0,1,1)$
See the big table for more information.
- none are K-stable
- every member is K‑polystable
- every member is K‑semistable
For this deformation family, every smooth 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 zero locus of the Futaki invariant contains a 2-dimensional family of Kähler classes including the anticanonical class.
For members with infinite automorphism group, the construction gives a 3-dimensional family near the anticanonical class.
This is the computation of Sektnan–Tipler.