In the example below, we take a smooth complete intersection $X\subset\mathbb{P}^5$ of three quadrics containing a conic $C\subset\mathbb{P}^5$. Then we calculate the map defined by the linear system $|2H+C|$, where $H$ is the hyperplane section class of $X$.
i1 : P5 = ZZ/65521[x_0..x_5]; |
i2 : C = ideal(x_1^2-x_0*x_2,x_3,x_4,x_5)
2
o2 = ideal (x - x x , x , x , x )
1 0 2 3 4 5
o2 : Ideal of P5
|
i3 : X = quotient ideal(-x_1^2+x_0*x_2-x_1*x_3+x_3^2-x_0*x_5+x_1*x_5+x_3*x_5,-x_0*x_3-x_1*x_3+x_2*x_4-x_3*x_4-x_4^2-x_1*x_5-x_2*x_5+x_5^2,-x_1^2+x_0*x_2+x_2*x_3+x_1*x_4-x_3*x_4-x_4*x_5); |
i4 : H = ideal random(1,X)
o4 = ideal(- 32646x - 28377x + 26433x - 29566x + 3783x + 26696x )
0 1 2 3 4 5
o4 : Ideal of X
|
i5 : D = new Tally from {H => 2,C => 1};
|
i6 : time phi = rationalMap D
-- used 0.0481745 seconds
o6 = -- rational map --
ZZ
source: subvariety of Proj(-----[x , x , x , x , x , x ]) defined by
65521 0 1 2 3 4 5
{
2 2
- x + x x - x x + x - x x + x x + x x ,
1 0 2 1 3 3 0 5 1 5 3 5
2 2
- x x - x x + x x - x x - x - x x - x x + x ,
0 3 1 3 2 4 3 4 4 1 5 2 5 5
2
- x + x x + x x + x x - x x - x x
1 0 2 2 3 1 4 3 4 4 5
}
ZZ
target: Proj(-----[y , y , y , y , y , y , y , y , y , y , y , y , y , y , y , y , y , y , y , y , y ])
65521 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
defining forms: {
x x x ,
0 4 5
2
x x ,
0 5
x x x ,
1 2 5
x x x ,
1 4 5
2
x x ,
1 5
2
x x ,
2 5
x x x ,
2 3 5
x x x ,
2 4 5
2
x x ,
2 5
2
x x ,
3 5
x x x ,
3 4 5
2
x x ,
3 5
2
x x ,
4 5
2
x x ,
4 5
3
x ,
5
2 2 2 2 2 2 2 3 2 2 2 2 2 2 2 2 3
x x + x x + x x x + x x - 3x x x - x x x - x x + x x - x x + 2x x - x - x x + 2x x x + 2x x x + x x + 2x x x - x x + 4x x x - 3x x x + x x + x x - 3x x - 2x x - 2x x + x ,
0 1 0 2 0 1 2 0 2 0 2 4 1 2 4 2 4 0 4 1 4 2 4 4 0 5 0 2 5 1 2 5 2 5 2 3 5 3 5 1 4 5 3 4 5 4 5 0 5 1 5 2 5 4 5 5
2 2 2 2 3 2 2 2 2 2 2 2 2 2 2 2 2 2 3
x x + x x x + x x + x x + 2x x + 32760x - 32760x x x - x x x - 32760x x + 32757x x x - 32760x x + 32760x x + x x - x x - 32760x x x - 32758x x x + 32760x x x + 32760x x + 2x x x - 3x x + 32758x x x + 7x x x + 32760x x x - 4x x x - 32760x x - 32758x x - 5x x - x x + 32759x x - 4x x - 32759x ,
0 2 0 1 2 0 2 1 2 2 3 3 0 2 4 1 2 4 2 4 2 3 4 3 4 2 4 3 4 0 5 0 1 5 0 2 5 1 2 5 2 5 2 3 5 3 5 0 4 5 1 4 5 2 4 5 3 4 5 4 5 0 5 1 5 2 5 3 5 4 5 5
2 2 3 2 2
x x x + x x + x x + x + x x x - 2x x + x x ,
0 1 2 0 2 1 2 2 1 2 4 2 4 2 4
2 2 2
- 2x x x + x x x - 2x x x + x x x + x x x - x x + x x + x x ,
0 2 5 0 4 5 1 4 5 2 4 5 3 4 5 4 5 1 5 4 5
2 2 2 2 2 2 2 2 3
- x x + 2x x x + 2x x x + x x + 2x x x - x x + 4x x x - 3x x x + x x + x x - 3x x - 2x x - 2x x + x ,
0 5 0 2 5 1 2 5 2 5 2 3 5 3 5 1 4 5 3 4 5 4 5 0 5 1 5 2 5 4 5 5
2 2
x x x + x x x + x x x + x x + x x x - 2x x x + x x
0 1 5 0 2 5 1 2 5 2 5 1 4 5 2 4 5 4 5
}
o6 : RationalMap (cubic rational map from surface in PP^5 to PP^20)
|
i7 : time ? image(phi,"F4")
-- used 0.922356 seconds
o7 = surface of degree 38 and sectional genus 20 in PP^20 cut out by 153
hypersurfaces of degree 2
|
See also the package Divisor, which provides general tools for working with divisors.