Calabi–Yau 3-folds

This section describes intrinsics that construct Calabi–Yau \(3\)-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 Buckley’s Riemann–Roch formula [Buckley, 2003] for polarised Calabi–Yau \(3\)-folds with canonical singularities.

CalabiYau(p1, p2, B): RngIntElt, RngIntElt, GRBskt -> GRCY

The Calabi–Yau \(3\)-fold \(X,A\) with \(h^0(X,A)=p_1\), \(h^0(X,2A)=p_2\) and basket of singularities \(B\).

FindN(X): GRCY -> RngIntElt, RngIntElt

MaximumN: RngIntElt                    Default: 100

The first nonnegative value, for the Calabi–Yau \(3\)-fold \(X\), of \(N_{C_i}\) (where \(C_i\) is the \(i\)th curve of the basket of \(X\)) together with the distance between successive values of \(N_{C_i}\). (The pair \(0,0\) is returned if no solutions below the parameter MaximumN are found).

FindN(p1, p2, B): RngIntElt, RngIntElt, GRBskt -> RngIntElt, RngIntElt

MaximumN: RngIntElt                    Default: 100

The first nonnegative value of \(N_{C_i}\) (where \(C_i\) is the \(i\)th curve of the basket \(B\) of \(3\)-fold points and curves and \(p_1\), \(p_2\) are the first two genera) together with the distance between successive values of \(N_{C_i}\). (The pair \(0,0\) is returned if no solutions below the parameter MaximumN are found).