

Fano 3folds database helpWhat is a Fano 3fold in the database?A Fano 3fold X is a normal projective 3fold with K_{X} ample. Unless otherwise mentioned, it is assumed also to have Qfactorial terminal singularities. (Some of the databases explicitly mention that they allow canonical singularities.) Usually a Fano 3fold is regarded as being polarised by its anticanonical class K_{X}. However, if K_{X} is divisible as a Weil divisor class, K_{X} = fA for maximal integer f, then the polarised pair X,A may be considered instead: this is called a Fano 3fold of index f. The Hilbert series of X,K_{X} (respectively X,A) can be computed using the IanoFletcherReid plurigenus formula (respectively Suzuki's RR formula). What is stored in the database?The Fano 3fold database is a list of Hilbert series. Each one is interpreted as a Fano 3fold embedded in weighted projective space (either with K_{X} or A as the degree 1 hyperplane). Such a description is called a candidate, since there is no guarantee that a Fano 3fold 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 databaseThere are standard names used in the Fano database. Let X be a Fano 3fold (with primitive divisor A if it is of index f > 1), and X in P(a_{0},...,a_{n}) be the corresponding embedding in wps.

