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.
Rationality
The family tables distinguish rationality from stable rationality. The qualifier “very general” is essential for stable irrationality: it describes the complement of a countable union of proper closed subsets of the moduli space, not every smooth member of the family. Rational families are stably rational. For the other families, the stable-irrationality statements follow from Hassett–Tschinkel, with the cubic-threefold case supplied by Engel–de Gaay Fortman–Schreieder.
Weyl groups and birational transformations
The displayed Weyl group describes symmetries of the KKMR chamber decomposition associated with the Fano threefold. See the Weyl-group overview and table for the definition, the 35 nontrivial cases, and sources.
Coregularity
The coregularity of a Fano variety is the dimension of its dual complex, subtracted from the dimension of the variety and then decreased by one. Thus coregularity zero is the extremal case in which a log Calabi–Yau boundary has a dual complex of maximal dimension. The first coregularity is the analogous invariant obtained using anticanonical boundaries only. The family statements use Avilov–Loginov–Przyjalkowski and the remaining existence results use Zhakupov.
Futaki invariant
The Futaki card concerns K-polystable members and varies the Kähler class over the Kähler cone. It records whether the invariant vanishes identically and, in the exceptional cases, a lower bound on the dimension of a family of zeroes containing the anticanonical class. The calculation is due to Sektnan–Tipler.
Atomic decompositions
For Picard-rank-one families, the semiorthogonal-decomposition card records both the naive atomic dimensions and a standard semiorthogonal decomposition from Böhning–Graf von Bothmer–Su'a.
Toric varieties
See the overview page for the list of toric Fano threefolds and rotatable diagrams of their polytopes.
As a zero section
For now, see the overview page.
K-stability
For now, see the overview page.