Identification
Fano variety 3-8
divisor from the linear system $|(\alpha\circ\pi_1)^*(\mathcal{O}_{\mathbb{P}^2}(1)\otimes\pi_2^*(\mathcal{O}_{\mathbb{P}^2}(2))|$ where $\pi_i\colon\mathrm{Bl}_1\mathbb{P}^2\times\mathbb{P}^2\to\mathrm{Bl}_1\mathbb{P}^2,\mathbb{P}^2$ are the projections, and $\alpha\colon\mathrm{Bl}_1\mathbb{P}^2\to\mathbb{P}^2$ is the blowup
- Picard rank
- 3 (others)
- $-\mathrm{K}_X^3$
- 24
- $\mathrm{h}^{1,2}(X)$
- 0
Hodge diamond and polyvector parallelogram
1
0 0
0 3 0
0 0 0 0
0 3 0
0 0
1
0 0
0 3 0
0 0 0 0
0 3 0
0 0
1
1
0 0*
0 3* 3
0 0 0 15
0 0 0
0 0
0
* indicates jumping of $\operatorname{Aut}^0$
its dimension takes on the values 0, 1
0 0*
0 3* 3
0 0 0 15
0 0 0
0 0
0
* indicates jumping of $\operatorname{Aut}^0$
its dimension takes on the values 0, 1
Anticanonical bundle
- index
- 1
- $\dim\mathrm{H}^0(X,\omega_X^\vee)$
- 15
- $-\mathrm{K}_X$ very ample?
- yes
- $-\mathrm{K}_X$ basepoint free?
- yes
- hyperelliptic
- no
- trigonal
- no
Birational geometry
Deformation theory
- number of moduli
- 3
- Bott vanishing
- does not hold
| $\mathrm{Aut}^0(X)$ | $\dim\mathrm{Aut}^0(X)$ | number of moduli |
|---|---|---|
| $\mathbb{G}_{\mathrm{m}}$ | 1 | 0 |
| $0$ | 0 | 3 |
Period sequence
Extremal contractions
Semiorthogonal decompositions
A full exceptional collection can be constructed using Orlov's blowup formula.
Structure of quantum cohomology
Zero section description
Fano 3-folds from homogeneous vector bundles over Grassmannians gives the following description(s):
- variety
- $\mathbb{P}^1 \times \mathbb{P}^2 \times \mathbb{P}^2$
- bundle
- $\mathcal{O}(0,1,2) \oplus \mathcal{O}(1,1,0)$
See the big table for more information.
K-stability
- general member is K‑stable but there exists one that is not
- general member is K‑polystable but there exists one that is not
- general member is K‑semistable