Fanography

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

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

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.

References

The original classification can be found in

The information on this website is based on

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

Acknowledgements

I would like to thank 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.