Given the ideal of a plane curve I=<g> and a pencil f0,f1 the ramification points are points in P1xP2 where t0f0+t1f1 and g are tangent, which can be define the minors of the 2x3 matrix of partials with respect to coordinates on P2. Saturating in possible base points of the pencil gives the branch points points, and the ramification is simple if the ideal b in kk[t0,t1] of the branchpoints is radical of the correct degree.
i1 : setRandomSeed "simple ramification"; |
i2 : kk=ZZ/10007; |
i3 : S=kk[x,y,z]; |
i4 : (I,Q,pencil) := randomCurveOfGenus9WithPencilOfDegree8 S; |
i5 : (ans,b)=simpleRamification(I,pencil)
32 31 30 2 29 3 28 4
o5 = (true, ideal(t + 2895t t + 713t t + 4920t t + 1764t t -
0 0 1 0 1 0 1 0 1
------------------------------------------------------------------------
27 5 26 6 25 7 24 8 23 9 22 10
3237t t + 3024t t + 3710t t - 1002t t + 3825t t - 1567t t +
0 1 0 1 0 1 0 1 0 1 0 1
------------------------------------------------------------------------
21 11 20 12 19 13 18 14 17 15
4459t t + 933t t + 3059t t - 1776t t + 4304t t -
0 1 0 1 0 1 0 1 0 1
------------------------------------------------------------------------
16 16 15 17 14 18 13 19 12 20
3434t t - 2087t t - 1189t t + 3602t t + 1192t t -
0 1 0 1 0 1 0 1 0 1
------------------------------------------------------------------------
11 21 10 22 9 23 8 24 7 25 6 26
3988t t - 1783t t + 1605t t + 2002t t - 2895t t + 1703t t
0 1 0 1 0 1 0 1 0 1 0 1
------------------------------------------------------------------------
5 27 4 28 3 29 2 30 31 32
- 1205t t - 2058t t - 4811t t - 3207t t - 4151t t + 2436t ))
0 1 0 1 0 1 0 1 0 1 1
o5 : Sequence
|
i6 : g=9,d=8 o6 = (9, 8) o6 : Sequence |
i7 : 2*g-2+2*d == degree b o7 = true |
i8 : cb = decompose b
13 12 11 2
o8 = {ideal(- 1230t + t ), ideal(- 2638t - 1706t t - 2260t t +
0 1 0 0 1 0 1
------------------------------------------------------------------------
10 3 9 4 8 5 7 6 6 7 5 8 4 9
645t t + 4296t t - 510t t - 1935t t - 380t t + 2018t t + 1986t t
0 1 0 1 0 1 0 1 0 1 0 1 0 1
------------------------------------------------------------------------
3 10 2 11 12 13 18 17
+ 4709t t - 655t t - 1509t t + t ), ideal(847t + 1503t t +
0 1 0 1 0 1 1 0 0 1
------------------------------------------------------------------------
16 2 15 3 14 4 13 5 12 6 11 7
2027t t + 2198t t - 2765t t - 332t t + 2232t t - 1843t t +
0 1 0 1 0 1 0 1 0 1 0 1
------------------------------------------------------------------------
10 8 9 9 8 10 7 11 6 12 5 13
2514t t + 3661t t + 3519t t + 1284t t + 4162t t + 2578t t +
0 1 0 1 0 1 0 1 0 1 0 1
------------------------------------------------------------------------
4 14 3 15 2 16 17 18
1769t t + 65t t - 2305t t - 4998t t + t )}
0 1 0 1 0 1 0 1 1
o8 : List
|
i9 : tally apply(cb,c->degree c)
o9 = Tally{1 => 1 }
13 => 1
18 => 1
o9 : Tally
|
i10 : I=ideal( z^2*(y^2-x^2)+x^4)
4 2 2 2 2
o10 = ideal(x - x z + y z )
o10 : Ideal of S
|
i11 : pencil=matrix{{y,z}}
o11 = | y z |
1 2
o11 : Matrix S <--- S
|
i12 : (ans,b)=simpleRamification(I,pencil)
2 2
o12 = (true, ideal(t - 4t ))
0 1
o12 : Sequence
|