Given the rational numbers or any finite field represented as a string, the procedure displays a string of M2-operations which calculate a unirational parametrization of the locus of lines in Q leading generically to numerical Godeaux surfaces with torsion group ℤ/5.
i1 : calculationOfTheUnirationalParametrizationOfTorsZ5Lines("QQ")
o1 = kk = value(s);
s = toString(1111);
(relLin,relPfaf,d1',d2,Ms) = setupGodeaux(kk,s);
lociE = lowerRankLociE(d1',relPfaf);
-- choose the P3s among the loci
lociEP3 = select(lociE,c-> codim c == 8);
Sa = getP11(relPfaf);
Q = sub(ideal relPfaf,Sa);
-- L gives the list of P1xP1's inside P3xP3's leading to Z/5-Godeaux surfaces:
L = lineConditionsTorsZ5(first lociEP3,last lociEP3,d1',relPfaf);
-- we choose the first P1xP1; the others are equivalent under the S4-operation
(comp0,comp1) = first L;
-- introduce coordinates for the P1xP1:
P1xP1 = kk[u_0..v_1,Degrees=>{2:{1,0},2:{0,1}}]
-- the generic point in the first P1:
genPoint0 =vars Sa % sub(comp0,Sa)
entriesGenPoint0 =flatten entries genPoint0;
-- the generic point in the second P1:
genPoint1 =vars Sa % sub(comp1,Sa);
entriesGenPoint1 =flatten entries genPoint1;
pos0 = positions(entriesGenPoint0, i-> i!=0);
pos1 = positions(entriesGenPoint1, i-> i!=0);
P1xP1xP1 = kk[u_0..v_1,x_0,x_1,Degrees => {2:{1,0,0},2:{0,1,0},{0,1,1},{1,0,1}}]
-- P1xP1xP1 is a P(O(1,0) ++ O(0,1)) -> P1xP1
-- the general line is given by:
phi = map(P1xP1xP1,Sa,apply(2,i-> entriesGenPoint0_(pos0_i) => x_0*u_i) | apply(2,i-> entriesGenPoint1_(pos1_i) => x_1*v_i) | apply(select(12,j-> not member(j,pos0|pos1)),i-> Sa_i => 0_P1xP1xP1))
-- the following two points span a general line leading to a Z/5-Godeaux surface:
phi0 = map(P1xP1,Sa,sub(diff(x_0,phi.matrix),P1xP1))
phi1 = map(P1xP1,Sa,sub(diff(x_1,phi.matrix),P1xP1))
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