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 |
i5 : ideal X
2 2
o5 = ideal (t - t t + t t , t t - t t + t t , t - t t + t t , t t -
5 4 6 2 7 4 5 3 6 1 7 4 3 5 0 7 2 4
------------------------------------------------------------------------
t t + t t , t t - t t + t t )
1 5 0 6 2 3 1 4 0 5
o5 : Ideal of frac(QQ[a..e])[t ..t ]
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 |
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 |
i8 : ideal oo
2
-d 2c -b - c + b*d -d c b
o8 = ideal (t + --t + --t + --t + ----------t , t + --t + -t + -t +
2 e 4 e 5 e 6 2 7 1 e 3 e 4 e 5
e
------------------------------------------------------------------------
2
-a - b*c + a*d -c 2b -a - b + a*c 2
--t + -----------t , t + --t + --t + --t + ----------t , t - t t
e 6 2 7 0 e 3 e 4 e 5 2 7 5 4 6
e e
------------------------------------------------------------------------
2
d -2c b c - b*d 2 d -c
+ -t t + ---t t + -t t + --------t , t t - t t + -t t + --t t +
e 4 7 e 5 7 e 6 7 2 7 4 5 3 6 e 3 7 e 4 7
e
------------------------------------------------------------------------
-b a b*c - a*d 2 2 c -2b a
--t t + -t t + ---------t , t - t t + -t t + ---t t + -t t +
e 5 7 e 6 7 2 7 4 3 5 e 3 7 e 4 7 e 5 7
e
------------------------------------------------------------------------
2
b - a*c 2
--------t )
2 7
e
o8 : Ideal of frac(QQ[a..e])[t ..t ]
0 7
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The object coneOfLines is a method function.