Fanography

About

Fano varieties are a special class of smooth projective varieties, taking up an important role in the classification of all smooth projective varieties. They are defined as the varieties for which the anticanonical bundle $\omega_X^\vee$ is ample. This ensures many nice properties for Fano varieties, such as

• the higher cohomology of the structure sheaf vanishes
• $\mathrm{Pic}(X)\cong\mathrm{H}^2(X,\mathbb{Z})$, so the Picard number $\rho$ equals the second Betti number
• there are only finitely many families of Fano varieties in each dimension

As they have a prominent role in the minimal model program and are interesting objects from the point of view of mirror symmetry, it makes sense to classify them.

dimension 1

There is only the projective line $\mathbb{P}^1$.

dimension 2

Fano varieties of dimension 2 (not to be confused with Fano surfaces) are also known as del Pezzo surfaces. They come in 10 families, and the complete list is:

• $\mathbb{P}^2$
• $\mathbb{P}^1\times\mathbb{P}^1$
• $\mathrm{Bl}_i\mathbb{P}^2$ the blowup of $\mathbb{P}^2$ in $i=1,\ldots,8$ points in general position

Whilst many interesting things can be said about del Pezzo surfaces, their classification is not too complicated.

This all changes in dimension 3, and the classification of Fano 3-folds is one of the main achievements of algebraic geometry in the last century. It was done in 2 steps:

1. by Iskovskikh, when $\mathrm{Pic}(X)\cong\mathbb{Z}$, see Fano threefolds I, II
2. by Mori–Mukai when $\rho(X)=B_2(X)\geq 2$, see Classification of Fano 3-folds with $B_2\geq 2$ (see also erratum)

The classification in dimension 4 is currently out of reach, but there are attempts at understanding it through mirror symmetry, see fanosearch.net.

By the way, let's assume we are working over $\mathbb{C}$ throughout.

How to cite?

If you use biblatex:

@online{fanography,
  author = {Belmans, Pieter},
  title = {Fanography},
  url = {https://fanography.info},
  year = {2024},
}

If you still use bibtex:

@misc{fanography,
  author = {Belmans, Pieter},
  title = {Fanography},
  howpublished = {\url{https://fanography.info}},
  year = {2024},
}

References

The original classification can be found in

• Iskovskikh: Fano threefolds I, II
• Mori–Mukai: Classification of Fano 3-folds with $B_2\geq 2$ (see also erratum)

The information on this website is based on

• the tables from Iskovskikh–Prokhorov: Fano varieties
• the tables from Cheltsov–Przyjalkowski–Shramov: Fano threefolds with infinite automorphism groups
• the tables from Kuznetsov–Prokhorov–Shramov: Hilbert schemes of lines and conics and automorphism groups of Fano threefolds
• the table from Codogni–Fanelli–Svaldi–Ttasin: Fano varieties in Mori fibre spaces

How to contribute

It is to be expected that there are typos, mistakes, or things that should just be better. If that is the case you can

• contact me (see below)
• create an issue on GitHub
• fix it yourself in data.yml and do a pull request on GitHub

Acknowledgements

I would like to thank Kristin DeVleming, Sergey Galkin, Ariyan Javanpeykar and Constantin Shramov for interesting discussions, and catching mistakes. A special thank you goes out to Alexander Kuznetsov for providing lots of information on moduli, and many interesting discussions. All remaining mistakes are due to my failures in typing, misinterpreting the literature or (let's hope not!) actual mistakes in the literature.

Contact

See pbelmans.ncag.info. Or just email me at pieterbelmans@gmail.com.