Construct a Prym canonical rational curve with g double points.
Step 1. Choosing an integer r and a prime p such that r represents an n primitive root of unity mod p. We then work over kk=ZZ/p and S=kk[x0,x1].
Step 2. Choose 2 times g points Pi,Qi randomly in P1(kk) which we indentify.
Step 3. Compute the canonical series of the singular curves and the multiplier A at the points, i.e. the ratio between the values of the sections at Pi and Qi.
Step 4. Change the ratio of the multipliers A at k of the double points by the root of unity, and compute the g-1 dimensional linear system of polynomials of degree 2g-2 on P1 satisfying these conditions.
Step 5. Computer the homogeneous ideal I of the image curve under the linear system
Return the basic objects L=(r,p,kk,S,PQ,A,s,T) and I
i1 : g=6;n:=3;k:=5; |
i4 : time (L,I)=randomPrymCanonicalNodalCurve(g,n,k);
-- used 0.103391 seconds
|
i5 : (r,p,kk,S,PQ,A,s,T)=L; |
i6 : L_0,L_1 o6 = (101, 10303) o6 : Sequence |
i7 : PQ
o7 = (| 4978 1473 -2166 -4917 1715 -3430 |, | -208 4654 2088 4600 -172
| -2812 2436 -4369 -2346 -1182 -2079 | | -1953 -238 -840 -4623 -3819
------------------------------------------------------------------------
416 |)
-1469 |
o7 : Sequence
|
i8 : A
o8 = | 2220 -1478 -4915 -4864 -3852 -4693 |
| 3885 -1441 -230 -2955 -1366 -2938 |
2 6
o8 : Matrix S <--- S
|
i9 : s;
1 5
o9 : Matrix S <--- S
|
i10 : betti res I
0 1 2 3
o10 = total: 1 10 15 6
0: 1 . . .
1: . . . .
2: . 10 15 6
o10 : BettiTally
|
i11 : numgens T==g-1, degree I== 2*g-2, genus(T/I)==g o11 = (true, true, true) o11 : Sequence |