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]); t = gens ring PP_K^4; phi = rationalMap {minors(2,matrix{{t_0,t_1,t_2},{t_1,t_2,t_3}}) + t_4};
o3 : MultirationalMap (rational map from PP^4 to PP^7)
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i4 : X = image phi;
o4 : ProjectiveVariety, 4-dimensional subvariety of PP^7
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i5 : ideal X
2 2
o5 = ideal (y - y y + y y , y y - y y + y y , y - y y + y y , y y -
5 4 6 2 7 4 5 3 6 1 7 4 3 5 0 7 2 4
------------------------------------------------------------------------
y y + y y , y y - y y + y y )
1 5 0 6 2 3 1 4 0 5
o5 : Ideal of frac(QQ[a..e])[y ..y ]
0 7
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i6 : p := projectiveVariety minors(2,(vars K)||(vars ring PP_K^4))
o6 = point of coordinates [a/e, b/e, c/e, d/e, 1]
o6 : ProjectiveVariety, a point in PP^4
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i7 : coneOfLines(X,phi p)
o7 = surface in PP^7 cut out by 6 hypersurfaces of degrees 1^3 2^3
o7 : ProjectiveVariety, surface in PP^7
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i8 : ideal oo
2
-d 2c -b - c + b*d -d c
o8 = ideal (y + --*y + --*y + --*y + ----------*y , y + --*y + -*y +
2 e 4 e 5 e 6 2 7 1 e 3 e 4
e
------------------------------------------------------------------------
b -a - b*c + a*d -c 2b -a
-*y + --*y + -----------*y , y + --*y + --*y + --*y +
e 5 e 6 2 7 0 e 3 e 4 e 5
e
------------------------------------------------------------------------
2 2
- b + a*c 2 d -2c b c - b*d 2
----------*y , y - y y + -*y y + ---*y y + -*y y + --------*y ,
2 7 5 4 6 e 4 7 e 5 7 e 6 7 2 7
e e
------------------------------------------------------------------------
d -c -b a b*c - a*d 2 2
y y - y y + -*y y + --*y y + --*y y + -*y y + ---------*y , y -
4 5 3 6 e 3 7 e 4 7 e 5 7 e 6 7 2 7 4
e
------------------------------------------------------------------------
2
c -2b a b - a*c 2
y y + -*y y + ---*y y + -*y y + --------*y )
3 5 e 3 7 e 4 7 e 5 7 2 7
e
o8 : Ideal of frac(QQ[a..e])[y ..y ]
0 7
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