option(none);

LIB "elim.lib";
LIB "solve.lib";
LIB "primdec.lib";

print("");
print("FIBER OF FERMAT MIRROR K3 (III)");
print(""); 
print("This program carries out the final step, allowing one to draw the");
print("fibre as ramified covering of C^*");

ring fourspace = (0,i),(u0,u1,u2,u3),dp; minpoly = i^2+1;

print("");

poly torus = u0*u1*u2*u3;
poly amap = (1/4)*(u0+u1+u2+u3);
poly bmap = (1/4)*(5/4*u0+i*u1-u2-i*u3); 
// poly cmap = z2; goes to C^* not C, so is highly non-generic

ideal solve1 = torus-1,amap,bmap;
ideal solve2 = elim1(solve1,u0*u1);
poly solve3 = solve2[1];

print("");
print("The embedding of the fibre in u2-u3 coordinates is:");
short = 0;
solve3;
print("");

print("Now we determine the u2-ramification values:");
ideal tangencies = solve3,diff(solve3,u3);
poly branch = elim1(tangencies,u3)[1];
branch;
option(prompt);