Fano 3-folds

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.

Example: Fano 3-Folds with the Same Basket (ex-109f4a)

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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 \(f,a,r-a\).

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Creation: \(f=1,2\) or \(\ge 3\)

Fano(f, B, g): RngIntElt, GRBskt, RngIntElt -> GRFano: A Fano 3-fold with Fano index \(f\ge 1\), Fano genus \(g\ge 0\), 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 \({{1}\over{r}}(a,b,c)\).

Fano(f, B): RngIntElt, GRBskt -> GRFano: A Fano \(3\)-fold with Fano index \(f\ge 3\) 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 \({{1}\over{r}}(a,b,c)\).

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 \(\ge 0\) 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 \(A(c_1(X)^2 - 3c_2(X))\) for the polarised Fano \(3\)-fold \(X,A\).

IsBogomolovUnstable(X): GRFano -> BoolElt: Return true if and only if the Bogomolov number \(A(c_1(X)^2 - 3c_2(X))\) 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 \(i\)th Fano \(3\)-fold in the Fano database \(D\).

Fano(D, f, i): DB, RngIntElt, RngIntElt -> GRFano: The \(i\)th Fano \(3\)-fold in the Fano database \(D\) that has Fano index \(f\).

Fano(D, f, Q, i): DB, SeqEnum, RngIntElt -> GRFano: The \(i\)th 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).

Fano 3-folds#

Creation: \(f=1,2\) or \(\ge 3\)#

A Preliminary Fano Database#

Fano 3-folds

Creation: \(f=1,2\) or \(\ge 3\)

A Preliminary Fano Database