A tool to visually study the geography of Fano 3-folds.


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


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.