Identification
Hodge diamond and polyvector parallelogram
1
0 0
0 2 0
0 3 3 0
0 2 0
0 0
1
0 0
0 2 0
0 3 3 0
0 2 0
0 0
1
1
0 0
0 9 3
0 0 0 13
0 0 0
0 0
0
0 0
0 9 3
0 0 0 13
0 0 0
0 0
0
Anticanonical bundle
- index
- 1
- $\dim\mathrm{H}^0(X,\omega_X^\vee)$
- 13
- $-\mathrm{K}_X$ very ample?
- yes
- $-\mathrm{K}_X$ basepoint free?
- yes
- hyperelliptic
- no
- trigonal
- no
Birational geometry
This variety is rational.
This variety is the blowup of
- 1-17, in a curve of genus 3
This variety is fibre-like, i.e. it can appear as the fibre of a Mori fibre space.
Deformation theory
- number of moduli
- 9
- Bott vanishing
- does not hold
$\mathrm{Aut}^0(X)$ | $\dim\mathrm{Aut}^0(X)$ | number of moduli |
---|---|---|
$0$ | 0 | 9 |
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
- $\mathbb{P}^3 \times \mathbb{P}^3$
- bundle
- $\mathcal{O}(1,1)^{\oplus 3}$
See the big table for more information.
K-stability
- general member is K‑stable
- general member is K‑polystable
- general member is K‑semistable
Hilbert schemes of curves
The Hilbert scheme of conics is the symmetric square of a genus 3 curve.
Its Hodge diamond is
1
3 3
3 10 3
3 3
1
3 3
3 10 3
3 3
1