The method as explained in HREF("http://arxiv.org","Unirationality of the Hurwitz space H9,8") constructs a triple of a irreducible nodal plane curve Γ of degree 8 and geometric genus 9 hence 12 double points, the ideal Q of 8 points on Γand a 1x2 matrix of quintics whish pass through the 12 double points of Γ and the points defined by Q. The moving intersection points of this pencil of quintics with Γ cut out the desired C.
i1 : kk=ZZ/10007; |
i2 : S=kk[x_0..x_2]; |
i3 : (I,Q,pencil)=randomCurveOfGenus9WithPencilOfDegree8 S; |
i4 : betti I
0 1
o4 = total: 1 1
0: 1 .
1: . .
2: . .
3: . .
4: . .
5: . .
6: . .
7: . 1
o4 : BettiTally
|
i5 : J = ideal pencil
5 4 3 2 2 3 4 5
o5 = ideal (- 1480x - 1582x x - 428x x - 1988x x + 792x x + 2728x -
0 0 1 0 1 0 1 0 1 1
------------------------------------------------------------------------
4 3 2 2 3 4 3 2
2241x x - 6x x x + 4446x x x - 942x x x - 2086x x - 1165x x +
0 2 0 1 2 0 1 2 0 1 2 1 2 0 2
------------------------------------------------------------------------
2 2 2 2 3 2 2 3 3 2 3
767x x x - 4234x x x - 2445x x - 2982x x + 3996x x x + 1993x x -
0 1 2 0 1 2 1 2 0 2 0 1 2 1 2
------------------------------------------------------------------------
4 4 5 5 4 3 2 2 3
4726x x - 720x x + 644x , - 4038x - 210x x + 4700x x - 1015x x +
0 2 1 2 2 0 0 1 0 1 0 1
------------------------------------------------------------------------
4 5 4 3 2 2 3
2698x x - 626x - 4252x x + 1654x x x + 1201x x x + 151x x x -
0 1 1 0 2 0 1 2 0 1 2 0 1 2
------------------------------------------------------------------------
4 3 2 2 2 2 2 3 2 2 3
3751x x - 3560x x - 4595x x x - 3014x x x - 2129x x + 398x x -
1 2 0 2 0 1 2 0 1 2 1 2 0 2
------------------------------------------------------------------------
3 2 3 4 4 5
4111x x x + 4070x x + 2344x x + 1444x x + 1561x )
0 1 2 1 2 0 2 1 2 2
o5 : Ideal of S
|
i6 : degree J == 25 o6 = true |
i7 : distinctPoints J o7 = true |
i8 : singI = saturate(ideal jacobian I +I); o8 : Ideal of S |
i9 : R = J:intersect(singI,Q); o9 : Ideal of S |
i10 : degree R == 5 o10 = true |
i11 : dim (R+I) == 0 o11 = true |
i12 : I
8 7 6 2 5 3 4 4 3 5
o12 = ideal(x + 3799x x + 2371x x - 3818x x + 396x x + 4485x x +
0 0 1 0 1 0 1 0 1 0 1
-----------------------------------------------------------------------
2 6 7 8 7 6 5 2
1056x x + 4065x x + 2276x + 1548x x - 3277x x x - 4824x x x +
0 1 0 1 1 0 2 0 1 2 0 1 2
-----------------------------------------------------------------------
4 3 3 4 2 5 6 7 6 2
4479x x x + 1286x x x - 1097x x x - 2170x x x - 1530x x + 1058x x
0 1 2 0 1 2 0 1 2 0 1 2 1 2 0 2
-----------------------------------------------------------------------
5 2 4 2 2 3 3 2 2 4 2 5 2
- 534x x x - 2433x x x - 4004x x x - 1342x x x - 1702x x x -
0 1 2 0 1 2 0 1 2 0 1 2 0 1 2
-----------------------------------------------------------------------
6 2 5 3 4 3 3 2 3 2 3 3 4 3
2059x x - 2423x x + 3417x x x + 476x x x + 581x x x - 2607x x x -
1 2 0 2 0 1 2 0 1 2 0 1 2 0 1 2
-----------------------------------------------------------------------
5 3 4 4 3 4 2 2 4 3 4 4 4
4706x x - 1225x x + 2213x x x + 4402x x x - 1681x x x + 2169x x -
1 2 0 2 0 1 2 0 1 2 0 1 2 1 2
-----------------------------------------------------------------------
3 5 2 5 2 5 3 5 2 6 6
4795x x - 3337x x x + 4773x x x - 3365x x + 905x x - 4733x x x +
0 2 0 1 2 0 1 2 1 2 0 2 0 1 2
-----------------------------------------------------------------------
2 6 7 7 8
3616x x + 4962x x - 4497x x - 3513x )
1 2 0 2 1 2 2
o12 : Ideal of S
|
i13 : transpose pencil
o13 = {-5} | -1480x_0^5-1582x_0^4x_1-428x_0^3x_1^2-1988x_0^2x_1^3+792x_0x_1^4
{-5} | -4038x_0^5-210x_0^4x_1+4700x_0^3x_1^2-1015x_0^2x_1^3+2698x_0x_1^
-----------------------------------------------------------------------
+2728x_1^5-2241x_0^4x_2-6x_0^3x_1x_2+4446x_0^2x_1^2x_2-942x_0x_1^3x_2-
4-626x_1^5-4252x_0^4x_2+1654x_0^3x_1x_2+1201x_0^2x_1^2x_2+151x_0x_1^3x
-----------------------------------------------------------------------
2086x_1^4x_2-1165x_0^3x_2^2+767x_0^2x_1x_2^2-4234x_0x_1^2x_2^2-
_2-3751x_1^4x_2-3560x_0^3x_2^2-4595x_0^2x_1x_2^2-3014x_0x_1^2x_
-----------------------------------------------------------------------
2445x_1^3x_2^2-2982x_0^2x_2^3+3996x_0x_1x_2^3+1993x_1^2x_2^3-4726x_0x_
2^2-2129x_1^3x_2^2+398x_0^2x_2^3-4111x_0x_1x_2^3+4070x_1^2x_2^3+2344x_
-----------------------------------------------------------------------
2^4-720x_1x_2^4+644x_2^5 |
0x_2^4+1444x_1x_2^4+1561x_2^5 |
2 1
o13 : Matrix S <--- S
|