Example GrdRng_gr-fano (H128E10)

We make two Fano 3-folds having the same basket but different genus.

> 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 + 1

In 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.

---

Creation: f=1, 2 or ≥3

Fano(f,B,g) : RngIntElt,GRBskt,RngIntElt -> GRFano

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).

Fano(f,B) : RngIntElt,GRBskt -> GRFano

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).

FanoIndex(X) : GRFano -> RngIntElt

The Fano index f of the Fano 3-fold X.

FanoGenus(X) : GRFano -> RngIntElt

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).

FanoBaseGenus(X) : GRFano -> RngIntElt

The smallest possible value for the Fano genus of the Fano 3-fold X.

BogomolovNumber(X) : GRFano -> FldRatElt

The intersection number A(c₁(X)² - 3c₂(X)) for the polarised Fano 3-fold X, A.

IsBogomolovUnstable(X) : GRFano -> BoolElt

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.

A Preliminary Fano Database

FanoDatabase() : -> DB

The database of Fano 3-folds.

Fano(D,i) : DB,RngIntElt -> GRFano

The ith Fano 3-fold in the Fano database D.

Fano(D,f,i) : DB,RngIntElt,RngIntElt -> GRFano

The ith Fano 3-fold in the Fano database D that has Fano index f.

Fano(D,f,Q,i) : DB,SeqEnum,RngIntElt -> GRFano

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).

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