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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]); 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)

Ways to use coneOfLines :

For the programmer

The object coneOfLines is a method function.