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SpecialFanoFourfolds :: coneOfLines

coneOfLines -- cone of lines on a subvariety passing through a point

Synopsis

Description

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 0.176063 seconds

o7 = surface in PP^7 cut out by 6 hypersurfaces of degrees 1^3 2^3 

o7 : ProjectiveVariety, surface in PP^7

Ways to use coneOfLines :

For the programmer

The object coneOfLines is a method function.