// This file contains code and output in SINGULAR verifying the claims made 
// about a pencil Pi discussed at the end of Section 5 (specifically, in 
// Prop. 5.3 under current numbering.)  

// Set up the coordinate chart away \tilde 0 and C, assuming Z_1=1.
// We work in characteristic 7.
> ring R=7,(x,y,Z,u,a),(dp(4),dp(1));
> poly curve = y2-x3+x2-x+1;
> poly alpha = 1;
> poly A = 2;
> poly invert = (x-alpha)*u-1;
> poly psi0 = (x-alpha)*Z^6+A*u;
> poly psi1 = (x-alpha)^2*Z^6+A^2*u^2;
> poly psi2 = (x-alpha)*Z^5+A*u*Z;
> poly psi3 = y*Z^5-A*y*u^2*Z;
> poly psi4 = Z^4+Z^2;
> poly psi5 = (x-alpha)*Z^4+A*u*Z^2;
> poly psi6 = Z^3;

// Define the pencil Pi (minus the element at infinity)
> poly s1 = psi0+psi3-psi5+psi6;
> poly s2 = psi0+psi1+psi2;
> poly pencil = a*s1+s2;

// Define locus of critical points, i.e., points where the rank of the differential
// is lower than expected.  This is the same as the vanishing of the largest minors
// in the differential.
> matrix singMat[2][3] = diff(curve,x), diff(curve,y), diff(curve,Z), 
. diff(pencil,x)+diff(pencil,u)*(-u^2), 
. diff(pencil,y), diff(pencil,Z),diff(pencil,u);
> ideal critPts = curve, pencil, invert, minor(singMat,2);
> critPts=stdfglm(critPts);


// This will show the critical points form a subscheme of dimension zero, one whose 
// coordinate ring is a vector space of dimension 64 over Q:
> dim(critPts);
0
> vdim(critPts);
64

// By the choice of our monomial ordering above, the first element of the ideal ‘critPts’
// will be a generator of the ideal sing \cap Q[a]; geometrically, its roots are the 
// values of a where corresponding element of the pencil is singular, i.e., they give 
// the critical values.  We see that there are 32 such values of a (in char. 7).
> critPts[1];
a32+3a31-a30+3a29-2a28+a27+3a26-a25-2a23-2a22-a21-2a20+3a19+3a18+a17-2a16+a15+a14+2a13+2a11+2a9+a8+2a7+3a6-a5+a4-2a3+3a+3


// We check that none of the critical points we have found lies over a 2-torsion 
// point of \tilde E:
> ideal i = critPts,y;
> i = stdfglm(i);
> dim(i);
-1

// By the preceding calculation, it follows that x-x0 and Z-Z0 form local parameters 
// of \tilde P at each critical point (where x=x0 and Z=Z0).  Hence an element of Pi 
// contains such a point as a singularity if (1) it passes through the point and (2) the
// x and Z derivatives of the point vanish there.  Looking for elements of \mathfrak d that
// are singular at the point, this gives a 3-by-7 matrix whose rank we must investigate; 
// if the rank of this matrix is less than 3, there is more than one element of 
// \mathfrak d which is singular at the point.

> matrix ratMapMat[3][4] =
. psi0, psi1, s1, s2,
. 2y*diff(psi0,x)+diff(psi0,y)*diff(curve,x)+2y*diff(psi0,u)*(-u^2), 
. 2y*diff(psi1,x)+diff(psi1,y)*diff(curve,x)+2y*diff(psi1,u)*(-u^2), 
. 2y*diff(s1,x)+diff(s1,y)*diff(curve,x)+2y*diff(s1,u)*(-u^2), 
. 2y*diff(s2,x)+diff(s2,y)*diff(curve,x)+2y*diff(s2,u)*(-u^2),
. diff(psi0,Z), diff(psi1,Z), diff(s1,Z), diff(s2,Z);
> ideal ratMapUndef = curve, pencil, invert, minor(ratMapMat,3); 
> ratMapUndef = stdfglm(ratMapUndef);
> dim(ratMapUndef);
0

// Having determined the ideal corresponding to those points at which more than one element
// of \mathfrak d is singular, we now compute that the set-theoretic intersection of its
// zero locus with the collection of critical points is empty.
> ideal intsect = critPts, ratMapUndef;
> intsect = stdfglm(intsect);
> dim(intsect);
-1