"Source: Text/Geometry/AlgSrf.text";
"Line: 3040";
"Date: Thu Sep 25 14:47:45 2025";
"Main: Fri Sep 26 14:29:19 2025";
// original file: Text/Geometry/AlgSrf.text, line: 3040
// Example: H126E15 ()
print "Example: H126E15";
ei := GetEchoInput();
SetEchoInput(true);
P<x,y,z,t> := ProjectiveSpace(GF(37),3);
S := Surface(P,
z^2*t^7-x^8*t-8*x^7*t*y-4*x^6*t^2*y-24*x^6*t*y^2-24*x^5*t^2*y^2-
4*x^6*y^3-32*x^5*t*y^3-52*x^4*t^2*y^3-24*x^5*y^4-32*x^4*t*y^4-
48*x^3*t^2*y^4-52*x^4*y^5-64*x^3*t*y^5-16*x^2*t^2*y^5-48*x^3*y^6-
96*x^2*t*y^6-16*x^2*y^7-64*x*t*y^7-16*t*y^8);
dsds := ResolveSingularSurface(S);
// one of the pairs of conjugate simple singularities has split over GF(37)
#dsds;
[NumberOfBlowUpDivisors(dsd) : dsd in dsds];
dsd := dsds[5];
ZetaFunctionOfResolutionFibre(dsd);
Factorisation(Denominator($1));
NumberOfPointsOnResolutionFibre(dsd);
A<a,b,c> := AffineSpace(GF(37),3);
S := Surface(A,-a^6-6*a^5*b-12*a^4*b^2-2*a^4*c-8*a^3*b^3-
     8*a^3*b*c-8*a^2*b^2*c+a^2*b^2-a^2*c^2+2*a*b^3-2*a*b*c^2+
     2*a*b*c+b^2*c+c^2);
dsds := ResolveSingularSurface(S);                                                                               
#dsds;
dsd := dsds[1];
ZetaFunctionOfResolutionFibre(dsd);
Factorisation(Denominator($1));
NumberOfPointsOnResolutionFibre(dsd);
SetEchoInput(ei);
