In the example below we compute the cone of lines passing through the generic point of a smooth del Pezzo fourfold in $\mathbb{P}^7$.
i1 : K := frac(QQ[a,b,c,d,e]); P4 = K[t_0..t_4]; phi = rationalMap(minors(2,matrix{{t_0,t_1,t_2},{t_1,t_2,t_3}}) + t_4);
o3 : RationalMap (quadratic rational map from PP^4 to PP^7)
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i4 : X = image phi
2 2
o4 = ideal (x - x x + x x , x x - x x + x x , x - x x + x x , x x - x x + x x , x x - x x + x x )
5 4 6 2 7 4 5 3 6 1 7 4 3 5 0 7 2 4 1 5 0 6 2 3 1 4 0 5
o4 : Ideal of frac(QQ[a..e])[x ..x ]
0 7
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i5 : p = phi minors(2,(vars K)||(vars P4))
2 2
-d -c -b -a c - b*d b*c - a*d b - a*c
o5 = ideal (x + --x , x + --x , x + --x , x + --x , x + --------x , x + ---------x , x + --------x )
6 e 7 5 e 7 4 e 7 3 e 7 2 2 7 1 2 7 0 2 7
e e e
o5 : Ideal of frac(QQ[a..e])[x ..x ]
0 7
|
i6 : time V = coneOfLines(X,p)
-- used 0.199091 seconds
2
-d 2c -b - c + b*d -d c b -a - b*c + a*d -c 2b -a
o6 = ideal (x + --x + --x + --x + ----------x , x + --x + -x + -x + --x + -----------x , x + --x + --x + --x +
2 e 4 e 5 e 6 2 7 1 e 3 e 4 e 5 e 6 2 7 0 e 3 e 4 e 5
e e
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2 2
- b + a*c 2 d -2c b c - b*d 2 d -c -b a b*c - a*d 2
----------x , x - x x + -x x + ---x x + -x x + --------x , x x - x x + -x x + --x x + --x x + -x x + ---------x ,
2 7 5 4 6 e 4 7 e 5 7 e 6 7 2 7 4 5 3 6 e 3 7 e 4 7 e 5 7 e 6 7 2 7
e e e
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2
2 c -2b a b - a*c 2
x - x x + -x x + ---x x + -x x + --------x )
4 3 5 e 3 7 e 4 7 e 5 7 2 7
e
o6 : Ideal of frac(QQ[a..e])[x ..x ]
0 7
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i7 : ? V o7 = cubic surface in PP^7 cut out by 6 hypersurfaces of degrees (1,1,1,2,2,2) |
The object coneOfLines is a method function.