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) |
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 |
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 : time coneOfLines(X,phi p) -- used 0.311861 seconds o7 = surface in PP^7 cut out by 6 hypersurfaces of degrees 1^3 2^3 o7 : ProjectiveVariety, surface in PP^7 |
The object coneOfLines is a method function.