Construct a canonical rational curve with g double points.
Step 1. Choose the prime p=32003. We then work over kk=ZZ/p and S=kk[x0,x1].
Step 2. Choose 2 times g points Pi,Qi randomly in PP1(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. Computer the homogeneous ideal I of the image curve under the linear system
Return the basic objects L=(kk,S,PQ,A,s,T) and I
i1 : g=6; |
i2 : time (L,I)=randomCanonicalNodalCurve(g);
-- used 0.092799 seconds
|
i3 : L_0
ZZ
o3 = -----
32003
o3 : QuotientRing
|
i4 : (kk,S,PQ,A,s,T)=L; |
i5 : PQ
o5 = (| 14750 -3404 12662 -11788 -2727 -11006 |, | -11631 -9701 9811
| -13290 2423 9395 -14782 -266 -425 | | 1312 -11718 -12012
------------------------------------------------------------------------
2398 3536 5497 |)
-3824 3060 -2566 |
o5 : Sequence
|
i6 : A
o6 = | -9946 -10088 15570 12621 9721 -3389 |
| 3739 -11922 -7503 11248 13519 515 |
2 6
o6 : Matrix S <--- S
|
i7 : s;
1 6
o7 : Matrix S <--- S
|
i8 : betti res I
0 1 2 3 4
o8 = total: 1 6 10 6 1
0: 1 . . . .
1: . 6 5 . .
2: . . 5 6 .
3: . . . . 1
o8 : BettiTally
|
i9 : numgens T==g, degree I== 2*g-2, genus(T/I)==g o9 = (true, true, true) o9 : Sequence |