Identification
Hodge diamond and polyvector parallelogram
1
0 0
0 1 0
0 0 0 0
0 1 0
0 0
1
0 0
0 1 0
0 0 0 0
0 1 0
0 0
1
1
0 0^{*}
0 6^{*} 3
0 0 0 14
0 0 0
0 0
0
^{*} indicates jumping of $\operatorname{Aut}^0$
its dimension takes on the values 0, 1, 3
0 0^{*}
0 6^{*} 3
0 0 0 14
0 0 0
0 0
0
^{*} indicates jumping of $\operatorname{Aut}^0$
its dimension takes on the values 0, 1, 3
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
This variety is rational.
This variety is primitive.
This variety is fibre-like, i.e. it can appear as the fibre of a Mori fibre space.
Deformation theory
- number of moduli
- 6
- Bott vanishing
- does not hold
$\mathrm{Aut}^0(X)$ | $\dim\mathrm{Aut}^0(X)$ | number of moduli |
---|---|---|
$\mathrm{PGL}_2$ | 3 | 0 |
$\mathbb{G}_{\mathrm{a}}$ | 1 | 0 |
$\mathbb{G}_{\mathrm{m}}$ | 1 | 1 |
$0$ | 0 | 6 |
Period sequence
Semiorthogonal decompositions
A full exceptional collection was constructed by Kuznetsov in 1996, see [MR1445274] .
Structure of quantum cohomology
Generic semisimplicity of:
- small quantum cohomology, proved by someone in at some point, see [?] , using
Zero section description
Fano 3-folds from homogeneous vector bundles over Grassmannians gives the following description(s):
- variety
- $\operatorname{Gr}(3,7)$
- bundle
- $(\bigwedge^2\mathcal{U}^\vee)^{\oplus 3}$
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
See the big table for more information.
Hilbert schemes of curves
The Hilbert scheme of conics is $\mathbb{P}^2$.
Its Hodge diamond is
1
0 0
0 1 0
0 0
1
0 0
0 1 0
0 0
1