Identification
Fano variety 3-6
blowup of 1-17 in the disjoint union of a line and an elliptic curve of degree 4
Alternative description:
- complete intersection of degree $(1,0,2)$ and $(0,1,1)$ in $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^3$
- Picard rank
- 3 (others)
- $-\mathrm{K}_X^3$
- 22
- $\mathrm{h}^{1,2}(X)$
- 1
Hodge diamond and polyvector parallelogram
1
0 0
0 3 0
0 1 1 0
0 3 0
0 0
1
0 0
0 3 0
0 1 1 0
0 3 0
0 0
1
1
0 0
0 5 2
0 0 0 14
0 0 0
0 0
0
0 0
0 5 2
0 0 0 14
0 0 0
0 0
0
Anticanonical bundle
- index
- 1
- $\dim\mathrm{H}^0(X,\omega_X^\vee)$
- 14
- $-\mathrm{K}_X$ very ample?
- yes
- $-\mathrm{K}_X$ basepoint free?
- yes
- hyperelliptic
- no
- trigonal
- no
Birational geometry
Deformation theory
- number of moduli
- 5
- Bott vanishing
- does not hold
$\mathrm{Aut}^0(X)$ | $\dim\mathrm{Aut}^0(X)$ | number of moduli |
---|---|---|
$0$ | 0 | 5 |
Period sequence
Extremal contractions
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
- $\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^3$
- bundle
- $\mathcal{O}(1,0,2) \oplus \mathcal{O}(0,1,1)$
See the big table for more information.