"Source: Text/Commut/PMod.text";
"Line: 3138";
"Date: Thu Sep 25 14:47:46 2025";
"Main: Fri Sep 26 14:29:19 2025";
// original file: Text/Commut/PMod.text, line: 3138
// Example: H119E17 ()
print "Example: H119E17";
ei := GetEchoInput();
SetEchoInput(true);
SetSeed(1);
R<x,y,z,t,u> := PolynomialRing(GF(17),5,"grevlex");
I := ideal<R |[
 2*x^3*z*t + 5*x^2*y*z*t + 14*x^2*z^2*t + x^3*z*u + 5*x^2*y*z*u +
  10*x^2*z^2*u + 8*x*y*z^2*u + 15*y^2*z^2*u + 4*x*z^3*u + 2*y*z^3*u +
  9*x^3*t*u + 14*x^2*y*t*u + 16*x^2*z*t*u + 10*x*y*z*t*u + 10*x*z^2*t*u +
  y*z^2*t*u + 13*x^3*u^2 + 14*x^2*y*u^2 + 11*x^2*z*u^2 + 15*x*y*z*u^2 +
  10*y^2*z*u^2 + 8*x*z^2*u^2 + 8*y*z^2*u^2 + x^2*t*u^2 + 11*x*y*t*u^2 +
  16*x*y*u^3 + 9*y^2*u^3 + 4*x*z*u^3 + 2*y*z*u^3 + 10*x*t*u^3 + y*t*u^3 +
  8*x*u^4 + 8*y*u^4,
 5*x^3*z*t + x^2*z^2*t + 5*x^3*z*u + 11*x^2*z^2*u + 15*x*y*z^2*u + 2*x*z^3*u
  + 14*x^3*t*u + 5*x^2*z*t*u + x*z^2*t*u + 14*x^3*u^2 + 10*x*y*z*u^2 +
  8*x*z^2*u^2 + 15*x^2*t*u^2 + 11*x^2*u^3 + 9*x*y*u^3 + 2*x*z*u^3 +
  x*t*u^3 + 8*x*u^4,
 14*x^3*z*t + x^2*y*z*t + 13*x^2*z^2*t + 7*x^2*z*t^2 + 3*x^3*z*u +
  16*x^2*y*z*u + 4*x^2*z^2*u + 6*x^3*t*u + 16*x^2*y*t*u + 9*x^2*z*t*u +
  9*x^2*t^2*u + 11*x^3*u^2 + x^2*y*u^2 + 14*x^2*z*u^2 + 2*x*z^2*u^2 +
  11*y*z^2*u^2 + 6*z^3*u^2 + 4*x^2*t*u^2 + 4*x*z*t*u^2 + 14*y*z*t*u^2 +
  6*z^2*t*u^2 + 15*x*t^2*u^2 + 10*z*t^2*u^2 + 3*x^2*u^3 + 11*x*z*u^3 +
  16*y*z*u^3 + 4*z^2*u^3 + 16*x*t*u^3 + 3*y*t*u^3 + 14*z*t*u^3 + 6*t^2*u^3
  + 13*x*u^4 + 7*y*u^4 + 16*z*u^4 + 11*t*u^4 + 10*u^5,
 15*x^3*z^2 + 12*x^2*y*z^2 + 3*x^2*z^3 + 12*x^3*z*u + 8*x^2*y*z*u +
  11*x^2*z^2*u + x^3*u^2 + 14*x^2*y*u^2 + 3*x^2*z*u^2 + 11*x^2*u^3,
 12*x^3*z^2 + 16*x^2*z^3 + 8*x^3*z*u + x^2*z^2*u + 14*x^3*u^2 + 16*x^2*z*u^2
  + x^2*u^3,
 2*x^3*y*z + 5*x^2*y^2*z + 14*x^2*y*z^2 + 13*x^3*y*u + 12*x^2*y^2*u +
  8*x^3*z*u + 4*x^2*y*z*u + 12*x^2*z^2*u + 3*x*y*z^2*u + 14*y^2*z^2*u +
  15*x*z^3*u + 3*y*z^3*u + 15*x^2*y*t*u + 4*x^2*z*t*u + 15*x*y*z*t*u +
  11*x*z^2*t*u + 10*y*z^2*t*u + 2*x^3*u^2 + 12*x^2*y*u^2 + 3*x^2*z*u^2 +
  14*x*y*z*u^2 + 13*x*z^2*u^2 + 10*y*z^2*u^2 + x^2*t*u^2 + 15*x*y*t*u^2 +
  16*y*z*t*u^2 + 12*x*y*u^3 + 3*y^2*u^3 + 15*x*z*u^3 + 4*y*z*u^3 +
  11*x*t*u^3 + 6*y*t*u^3 + 13*x*u^4 + 14*y*u^4,
 5*x^3*y*z + x^2*y*z^2 + 10*x^2*z^2*t + 4*x^4*u + 12*x^3*y*u + 5*x^3*z*u +
  12*x^2*y*z*u + 14*x*y*z^2*u + 3*x*z^3*u + 15*x^3*t*u + 16*x^2*z*t*u +
  10*x*z^2*t*u + 11*x^3*u^2 + 4*x^2*y*u^2 + 13*x^2*z*u^2 + 10*x*z^2*u^2 +
  4*x^2*t*u^2 + 16*x*z*t*u^2 + 13*x^2*u^3 + 3*x*y*u^3 + 4*x*z*u^3 +
  6*x*t*u^3 + 14*x*u^4,
 10*x^2*z^3 + 8*x^2*z^2*u + 5*x^2*z*u^2 + 11*x^2*u^3,
 16*x^3*z^2 + 12*x^2*y*z^2 + 7*x^2*z^3 + 9*x*y*z^3 + 2*y^2*z^3 + 13*x*z^4 +
  15*y*z^4 + 13*x^3*z*t + 12*x^2*y*z*t + 7*x^2*z^2*t + 7*x*y*z^2*t +
  7*x*z^3*t + 16*y*z^3*t + 4*x^3*z*u + 3*x^2*y*z*u + 6*x^2*z^2*u +
  2*x*y*z^2*u + 7*y^2*z^2*u + 9*x*z^3*u + 9*y*z^3*u + 16*x^3*t*u +
  3*x^2*y*t*u + 13*x^2*z*t*u + 6*x*y*z*t*u + x*y*z*u^2 + 8*y^2*z*u^2 +
  13*x*z^2*u^2 + 15*y*z^2*u^2 + 6*x^2*t*u^2 + 7*x*z*t*u^2 + 16*y*z*t*u^2 +
  9*x*z*u^3 + 9*y*z*u^3,
 12*x^3*z^2 + 6*x^2*z^3 + 2*x*y*z^3 + 15*x*z^4 + 12*x^3*z*t + 11*x^2*z^2*t +
  16*x*z^3*t + 3*x^3*z*u + 7*x*y*z^2*u + 9*x*z^3*u + 3*x^3*t*u +
  3*x^2*z*t*u + 6*x^2*z*u^2 + 8*x*y*z*u^2 + 15*x*z^2*u^2 + 16*x^2*t*u^2 +
  16*x*z*t*u^2 + 9*x*z*u^3,
 x^3*y*z + 5*x^2*y^2*z + 10*x^2*y*z^2 + 8*x*y^2*z^2 + 15*y^3*z^2 + 4*x*y*z^3
  + 2*y^2*z^3 + 13*x^3*y*t + 2*x^2*y^2*t + 9*x^3*z*t + 12*x^2*y*z*t +
  10*x*y^2*z*t + 5*x^2*z^2*t + 7*x*y*z^2*t + 4*y^2*z^2*t + 2*x*z^3*t +
  14*y*z^3*t + 2*x^2*y*t^2 + 13*x^2*z*t^2 + 2*x*y*z*t^2 + 6*x*z^2*t^2 +
  7*y*z^2*t^2 + 13*x^3*y*u + 14*x^2*y^2*u + 11*x^2*y*z*u + 15*x*y^2*z*u +
  10*y^3*z*u + 8*x*y*z^2*u + 8*y^2*z^2*u + 15*x^3*t*u + 6*x^2*y*t*u +
  11*x*y^2*t*u + 14*x^2*z*t*u + 3*x*y*z*t*u + 4*x*z^2*t*u + 7*y*z^2*t*u +
  16*x^2*t^2*u + 2*x*y*t^2*u + y*z*t^2*u + 16*x*y^2*u^2 + 9*y^3*u^2 +
  4*x*y*z*u^2 + 2*y^2*z*u^2 + 15*x*y*t*u^2 + 15*y^2*t*u^2 + 2*x*z*t*u^2 +
  13*y*z*t*u^2 + 6*x*t^2*u^2 + 11*y*t^2*u^2 + 8*x*y*u^3 + 8*y^2*u^3 +
  4*x*t*u^3 + 3*y*t*u^3,
 5*x^3*y*z + 11*x^2*y*z^2 + 15*x*y^2*z^2 + 2*x*y*z^3 + 13*x^4*t + 2*x^3*y*t +
  7*x^3*z*t + 6*x^2*y*z*t + 16*x^2*z^2*t + 4*x*y*z^2*t + 14*x*z^3*t +
  2*x^3*t^2 + 9*x^2*z*t^2 + 7*x*z^2*t^2 + 14*x^3*y*u + 5*x^3*z*u +
  4*x^2*y*z*u + 10*x*y^2*z*u + 12*x^2*z^2*u + 8*x*y*z^2*u + 15*x*z^3*u +
  6*y*z^3*u + 11*z^4*u + 16*x^3*t*u + 15*x^2*y*t*u + 13*x^2*z*t*u +
  3*x*z^2*t*u + 3*y*z^2*t*u + 11*z^3*t*u + 13*x^2*t^2*u + 3*x*z*t^2*u +
  7*z^2*t^2*u + 7*x^3*u^2 + 7*x^2*y*u^2 + 9*x*y^2*u^2 + x^2*z*u^2 +
  2*x*y*z*u^2 + 6*x*z^2*u^2 + y*z^2*u^2 + 13*z^3*u^2 + 12*x^2*t*u^2 +
  15*x*y*t*u^2 + 14*x*z*t*u^2 + 14*y*z*t*u^2 + 3*z^2*t*u^2 + 11*x*t^2*u^2
  + 11*z*t^2*u^2 + 9*x^2*u^3 + 8*x*y*u^3 + 4*x*z*u^3 + 10*y*z*u^3 +
  z^2*u^3 + 3*x*t*u^3 + 6*z*t*u^3 + 7*z*u^4,
 4*x^4*z + 3*x^3*y*z + 10*x^3*z^2 + x^2*z^3 + 14*x*y*z^3 + 3*x*z^4 +
  15*x^3*z*t + 8*x^2*z^2*t + 10*x*z^3*t + 14*x^3*y*u + 15*x^3*z*u +
  11*x^2*z^2*u + 10*x*z^3*u + 16*x^2*z*t*u + 16*x*z^2*t*u + 6*x^3*u^2 +
  14*x^2*z*u^2 + 3*x*y*z*u^2 + 4*x*z^2*u^2 + 6*x^2*t*u^2 + 6*x*z*t*u^2 +
  15*x^2*u^3 + 14*x*z*u^3,
 5*x^3*z^2 + 4*x^2*y*z^2 + 12*x^2*z^3 + 15*x*z^4 + 6*y*z^4 + 11*z^5 +
  14*x^3*z*t + x^2*y*z*t + 7*x^2*z^2*t + 13*x*z^3*t + 3*y*z^3*t + 11*z^4*t
  + 12*x^2*z*t^2 + 2*x*z^2*t^2 + 7*z^3*t^2 + 7*x^3*z*u + 13*x^2*y*z*u +
  x^2*z^2*u + 6*x*z^3*u + y*z^3*u + 13*z^4*u + 6*x^3*t*u + 16*x^2*y*t*u +
  9*x^2*z*t*u + x*z^2*t*u + 14*y*z^2*t*u + 3*z^3*t*u + 6*x^2*t^2*u +
  11*z^2*t^2*u + 9*x^2*z*u^2 + 4*x*z^2*u^2 + 10*y*z^2*u^2 + z^3*u^2 +
  15*x^2*t*u^2 + 6*z^2*t*u^2 + 7*z^2*u^3,
 13*x^5 + 2*x^4*y + 7*x^4*z + 6*x^3*y*z + 16*x^3*z^2 + 4*x^2*y*z^2 +
  14*x^2*z^3 + 2*x^4*t + 9*x^3*z*t + 7*x^2*z^2*t + 16*x^4*u + 15*x^3*y*u +
  x^3*z*u + 11*x^2*z^2*u + 3*x*y*z^2*u + 14*x*z^3*u + 13*x^3*t*u +
  3*x^2*z*t*u + 7*x*z^2*t*u + 14*x^3*u^2 + 15*x^2*y*u^2 + 6*x^2*z*u^2 +
  14*x*y*z*u^2 + 10*x*z^2*u^2 + 11*x^2*t*u^2 + 11*x*z*t*u^2 + 3*x*z*u^3]>;
X := Scheme(Proj(R),I); // define the surface
// The structure sheaf O_X of X is represented by graded module R/I
O_X := GradedModule(I); // R/I
CohomologyDimension(O_X,0,0);  // dim H^0(O_X)
assert $1 eq 1;
// get the geometric genus and irregularity
time CohomologyDimension(O_X,2,0); // dim H^2(O_X) = g
assert $1 eq 0;
time CohomologyDimension(O_X,1,0); // dim H^1(O_X) = q
assert $1 eq 0;
// => p_a(X)=0. Verify this.
ArithmeticGenus(X);
assert $1 eq 0;
K_X := Ext(2,O_X,RModule(R,[5]));
K_X;
CohomologyDimension(K_X,0,0);
assert $1 eq 0;
[CohomologyDimension(K_X,0,i) eq CohomologyDimension(O_X,2,-i) :
   i in [-1..5]];
SetEchoInput(ei);
