Statistics
| Picard rank $\rho$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | total |
|---|---|---|---|---|---|---|---|---|---|---|---|
| number of deformation families | 17 | 36 | 31 | 13 | 3 | 1 | 1 | 1 | 1 | 1 | 105 |
| number of primitive deformation families | 17 | 9 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 30 |
| number of rational deformation families | 8 | 29 | 30 | 13 | 3 | 1 | 1 | 1 | 1 | 1 | 88 |
| number of unirational deformation families | 14 | 35 | 31 | 13 | 3 | 1 | 1 | 1 | 1 | 1 | 101 |
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Terminology and notation
We would like to discuss the terminology which appears in the classification, but we haven't written this yet. If you would like to contribute, please get in touch!
The (anti)canonical bundle
- hyperelliptic
The linear system $|-\mathrm{K}_X|$ is basepointfree, but not very ample, and it defines a morphism of degree 2 onto its image.
- del Pezzo varieties
These are pairs $(X,H)$ of a smooth projective variety $X$ and an ample divisor $H$ such that $-\mathrm{K}_X=(\dim X-1)H$. So for Fano 3-folds these have (co)index 2 and $H$ is the generator of the Picard group, or $X=\mathbb{P}^3$ and $H$ is twice the generator of the Picard group.
For more information, see the overview page.
Automorphism groups
The connected component of the identity in the whole automorphism group is described. The reason for this is that the whole automorphism group jumps quite a bit, already for del Pezzo surfaces (of low degree).
At some point we should explain the notation which is used for automorphism groups, but for now one is referred to the notation overview on pages 4 and 5 of Cheltsov–Przyjalkowski–Shramov: Fano threefolds with infinite automorphism groups.
Toric varieties
For now, see the overview page.
As a zero section
For now, see the overview page.
K-stability
For now, see the overview page.