getSections -- computes a matrix defining the canonical embedding

Constructs a matrix s of sections that defines an embedding PP¹->PP^g-1 such that the image is a k-gonal g-nodal canonical curve. The construction of the sections is based on the construction implemented in the M2 package NodalCurves.

Step 1. We search f:S²(-k)→S such that det(f|pt_i) has at least two linear or one quadratic factor for g distinct points pt_i in an affine chart of PP¹. This part is done by the function listOfFactors.

Step 2.We build g quadrics q_i: if det(f|pt_i) has two linear factors then we commpute the two points P_i and Q_i as the vanishing loci of the linear factors and define q_i :=det(P_i | (x₀,x₁)^t) * det(Q_i | (x₀,x₁)^t). If det(f|pt_i) already has a quadratic factor we can definie q_i to be this factor.

Step 3. We compute a basis of H⁰(C,ω_C) by {s_i :=∏^g_j≠i,j=1q_i | i=1,...,g } and multiply the matrix s=(s₁,...,s_g) with a general matrix M∈GL(g,kk) to obtain more general sections.

i1 : (k,g,n)=(4,8,100);

i2 : (p,kk,S)=getFieldAndRing(n);

i3 : time s=sub(getSections(k,g,n),S);
     -- used 0.110574 seconds

             1       8
o3 : Matrix S  <--- S

i4 : T=kk[t_0..t_(g-1)];

i5 : time I=ideal mingens ker map(S,T,s);
     -- used 0.476314 seconds

o5 : Ideal of T

i6 : time betti res I
     -- used 2.56267 seconds

            0  1  2  3  4  5 6
o6 = total: 1 15 39 50 39 15 1
         0: 1  .  .  .  .  . .
         1: . 15 35 25  4  . .
         2: .  .  4 25 35 15 .
         3: .  .  .  .  .  . 1

o6 : BettiTally