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) |
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 |
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 ---------------------------------------------------------------------------------------------------------------------------- 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 ---------------------------------------------------------------------------------------------------------------------------- 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 |
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.