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 |