Fano 3-folds database help

What is a Fano 3-fold in the database?

A Fano 3-fold X is a normal projective 3-fold with -KX ample. Unless otherwise mentioned, it is assumed also to have Q-factorial terminal singularities. (Some of the databases explicitly mention that they allow canonical singularities.) Usually a Fano 3-fold is regarded as being polarised by its anticanonical class -KX. However, if -KX is divisible as a Weil divisor class, -KX = fA for maximal integer f, then the polarised pair X,A may be considered instead: this is called a Fano 3-fold of index f.

The Hilbert series of X,-KX (respectively X,A) can be computed using the Iano-Fletcher--Reid plurigenus formula (respectively Suzuki's RR formula).

What is stored in the database?

The Fano 3-fold database is a list of Hilbert series. Each one is interpreted as a Fano 3-fold embedded in weighted projective space (either with -KX or A as the degree 1 hyperplane). Such a description is called a candidate, since there is no guarantee that a Fano 3-fold with the given Hilbert series exists, or, if it does, that it can be embedded as indicated. In practice, all candidates in codimension at most 4, and many in other low codimensions, have been confirmed.

Glossary of terms for the Fano database

There are standard names used in the Fano database. Let X be a Fano 3-fold (with primitive divisor A if it is of index f > 1), and X in P(a0,...,an) be the corresponding embedding in wps.

  • Basket: A collection of types of quotient singularities 1/r(f,a,-a) (sometimes denoted [r,a] if the index f is clear) that make the same contribution to RR as the actual singularities of X. The singularities of X are often exactly those listed in the basket.
  • Codimension: The codimension of X in its embeddeding in wps.
  • Degree: The rational number (-KX)3
  • Genus: dim H0(X,-KX) - 2 (or h0 - 1 if f > 1)
  • Index: The number of X in the database.
  • Weights: The weights the ambient wps of X.
  • Kawamata number: The sum of r - (1/r) over the singularities 1/r(f,a,-a) of the basket. This number is bounded above by 24, one of the ingredients in Kawamata's proof of boundedness of Fano 3-folds.
  • Bogomolov number: f2A3 - 3Ac2(X), a number that must be nonpositive if the saturated sheaf of 1-differentials of X is to be stable. This stability is not imposed as a condition on Fano 3-folds, although no unstable candidates have been constructed as varieties and Prokhorov has proved that Suzuki's unstable index 10 candidate does not exist as a variety.