This section describes intrinsics that construct Fano 3-folds. It also describes a few intrinsics that can be used to study them, but see Section Generic Polarised Varieties for the general intrinsics that apply to all polarised varieties.
The calculations are based on Suzuki's Riemann--Roch formula [Suz] for polarised Fano 3-folds with terminal singularities.
> X := Fano(2,MakeBasket([[3,1,2,2]]),2); > X; Fano 3-fold X,A of Fano index 2, Fano genus 2, in codimension 1 with data Weights: [ 1, 1, 2, 3, 5 ] Basket: 1/3(1,2,2) Degrees: A^3 = 1/3, (1/12)Ac_2(X) = 8/9 Numerator: -t^10 + 1 > FanoGenus(X); 2 > FanoBaseGenus(X); 2 > Fano(2,MakeBasket([[3,1,2,2]]),3); Fano 3-fold X,A of Fano index 2, Fano genus 3, in codimension 2 with data Weights: [ 1, 1, 1, 2, 2, 3 ] Basket: 1/3(1,2,2) Degrees: A^3 = 4/3, (1/12)Ac_2(X) = 8/9 Numerator: t^8 - 2*t^4 + 1In this example, the smallest possible genus---the Fano base genus---is 2, and an error will be reported if a smaller value is requested.
Note that the singularities used must be polarised by fA, where f is the Fano index: in practice, this means their index r must be coprime to f and their weights must be of the form f, a, r - a.
A Fano 3-fold with Fano index f≥1, Fano genus g≥0 and basket of singularities B. The singularities must be terminal singularities. The basket can also be presented in raw sequence format: in this case, B is a sequence containing terms such as [r, a, b, c] which denotes the singularity oneover(r)(a, b, c).
A Fano 3-fold with Fano index f≥3 and basket of singularities B. The singularities must be terminal singularities. The basket can also be presented in raw sequence format: in this case, B is a sequence containing terms such as [r, a, b, c] which denotes the singularity oneover(r)(a, b, c).
The Fano index f of the Fano 3-fold X.
The Fano genus of the Fano 3-fold X, an integer ≥0 equal to the dimension of the space of sections of the polarising divisor. (The term genus often refers to two less that this number).
The smallest possible value for the Fano genus of the Fano 3-fold X.
The intersection number A(c1(X)2 - 3c2(X)) for the polarised Fano 3-fold X, A.
Return true if and only if the Bogomolov number A(c1(X)2 - 3c2(X)) for the polarised Fano 3-fold X, A is strictly positive.
The database of Fano 3-folds.
The ith Fano 3-fold in the Fano database D.
The ith Fano 3-fold in the Fano database D that has Fano index f.
The ith Fano 3-fold in the Fano database D that has Fano index f and initial plurigenera as specified by the sequence Q (up to the first four plurigenera).