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Cremona :: kernelComponent

kernelComponent -- about the image of a rational map

Synopsis

Description

Assume, for simplicity, that J=0, and let φ:K[y0,...,ym] →K[x0,...,xn] be a ring map representing a rational map Φ: ℙn=Proj(K[x0,...,xn]) ---> ℙm=Proj(K[y0,...,ym]). Then kernelComponent(phi,d) returns the ideal generated by a basis of the vector space H0(ℙm,I(d)), where I denotes the ideal sheaf of the image of Φ.

This is equivalent to ideal image basis(d,kernel phi), but a more direct algorithm is used. Indeed, taking a generic homogeneous degree d polynomial G(y0,...,ym) and substituting the yi's with the Fi's, where Fi=Fi(x0,...,xn)=φ(yi), we obtain a homogeneous polynomial that vanishes identically if and only if G lies in the kernel of phi; thus, the problem is reduced to resolving a homogeneous linear system in the coefficients of G.

i1 : -- A special birational transformation of P^8 into a complete intersection of three quadrics in P^11
     K=QQ; ringP8=K[x_0..x_8]; ringP11=K[y_0..y_11];
i4 : phi=map(ringP8,ringP11,{-5*x_0*x_3+x_2*x_4+x_3*x_4+35*x_0*x_5-7*x_2*x_5+x_3*x_5-x_4*x_5-49*x_5^2-5*x_0*x_6+2*x_2*x_6-x_4*x_6+27*x_5*x_6-4*x_6^2+x_4*x_7-7*x_5*x_7+2*x_6*x_7-2*x_4*x_8+14*x_5*x_8-4*x_6*x_8,-x_1*x_2-6*x_1*x_5-5*x_0*x_6+2*x_1*x_6+x_4*x_6+x_5*x_6-5*x_0*x_7-x_1*x_7+2*x_2*x_7+7*x_5*x_7-2*x_6*x_7+2*x_1*x_8-3*x_7*x_8,-25*x_0^2+9*x_0*x_2+10*x_0*x_4-2*x_2*x_4-x_4^2+29*x_0*x_5-x_2*x_5-7*x_4*x_5-13*x_0*x_6+3*x_4*x_6+x_5*x_6-x_0*x_7+2*x_2*x_7-x_4*x_7+7*x_5*x_7-2*x_6*x_7-8*x_0*x_8+2*x_4*x_8-3*x_7*x_8,x_2*x_4+x_3*x_4+x_4^2+7*x_2*x_5-9*x_4*x_5+12*x_5*x_6-4*x_6^2+2*x_3*x_7+2*x_4*x_7-14*x_5*x_7+4*x_6*x_7+x_3*x_8-x_4*x_8-14*x_5*x_8+x_6*x_8,-5*x_0*x_4+x_2*x_4-7*x_2*x_5+8*x_4*x_5-5*x_0*x_6+2*x_2*x_6-x_4*x_6+x_5*x_6-x_4*x_7+7*x_5*x_7-2*x_6*x_7-x_4*x_8+7*x_5*x_8-2*x_6*x_8,x_0*x_4+x_4^2-7*x_1*x_5-8*x_4*x_5+x_0*x_6+x_1*x_6+2*x_4*x_6-x_5*x_6+x_4*x_7-7*x_5*x_7+2*x_6*x_7+x_4*x_8-7*x_5*x_8+2*x_6*x_8,x_2*x_3+x_4^2-8*x_4*x_5+x_4*x_6+6*x_5*x_6-2*x_6^2+x_3*x_7+x_4*x_7-7*x_5*x_7+2*x_6*x_7+x_4*x_8-7*x_5*x_8+2*x_6*x_8,x_1*x_3-7*x_1*x_5+x_1*x_6+x_4*x_6-7*x_5*x_6+2*x_6^2-x_3*x_7,-4*x_0*x_3+x_3*x_4-x_4^2-7*x_0*x_5+8*x_4*x_5+x_0*x_6-x_4*x_6-6*x_5*x_6+2*x_6^2-x_3*x_7-x_4*x_7+7*x_5*x_7-2*x_6*x_7-x_4*x_8+7*x_5*x_8-2*x_6*x_8,-5*x_0*x_2+2*x_2^2+x_2*x_4-x_4^2-x_2*x_5+8*x_4*x_5-10*x_0*x_6+2*x_5*x_6+2*x_2*x_7-2*x_4*x_7+14*x_5*x_7-4*x_6*x_7+5*x_0*x_8-3*x_2*x_8-2*x_4*x_8+7*x_5*x_8-2*x_6*x_8-3*x_7*x_8,-5*x_0*x_1+x_1*x_2+x_1*x_4-4*x_0*x_6-x_1*x_6+x_4*x_6+x_0*x_7,x_0*x_2-x_1*x_2+5*x_0*x_4+x_1*x_4-14*x_1*x_5-x_2*x_5-8*x_4*x_5-8*x_0*x_6+2*x_1*x_6+4*x_4*x_6+2*x_2*x_7+4*x_0*x_8+3*x_1*x_8-7*x_5*x_8+2*x_6*x_8-3*x_7*x_8})

                                                                                   2                                     2                                                                                                                                                            2                             2                                                                                                                               2                              2                                                                                                                                                                                     2                                                                                                     2                            2                                                                                              2                           2                                           2                                                                   2           2
o4 = map(ringP8,ringP11,{- 5x x  + x x  + x x  + 35x x  - 7x x  + x x  - x x  - 49x  - 5x x  + 2x x  - x x  + 27x x  - 4x  + x x  - 7x x  + 2x x  - 2x x  + 14x x  - 4x x , - x x  - 6x x  - 5x x  + 2x x  + x x  + x x  - 5x x  - x x  + 2x x  + 7x x  - 2x x  + 2x x  - 3x x , - 25x  + 9x x  + 10x x  - 2x x  - x  + 29x x  - x x  - 7x x  - 13x x  + 3x x  + x x  - x x  + 2x x  - x x  + 7x x  - 2x x  - 8x x  + 2x x  - 3x x , x x  + x x  + x  + 7x x  - 9x x  + 12x x  - 4x  + 2x x  + 2x x  - 14x x  + 4x x  + x x  - x x  - 14x x  + x x , - 5x x  + x x  - 7x x  + 8x x  - 5x x  + 2x x  - x x  + x x  - x x  + 7x x  - 2x x  - x x  + 7x x  - 2x x , x x  + x  - 7x x  - 8x x  + x x  + x x  + 2x x  - x x  + x x  - 7x x  + 2x x  + x x  - 7x x  + 2x x , x x  + x  - 8x x  + x x  + 6x x  - 2x  + x x  + x x  - 7x x  + 2x x  + x x  - 7x x  + 2x x , x x  - 7x x  + x x  + x x  - 7x x  + 2x  - x x , - 4x x  + x x  - x  - 7x x  + 8x x  + x x  - x x  - 6x x  + 2x  - x x  - x x  + 7x x  - 2x x  - x x  + 7x x  - 2x x , - 5x x  + 2x  + x x  - x  - x x  + 8x x  - 10x x  + 2x x  + 2x x  - 2x x  + 14x x  - 4x x  + 5x x  - 3x x  - 2x x  + 7x x  - 2x x  - 3x x , - 5x x  + x x  + x x  - 4x x  - x x  + x x  + x x , x x  - x x  + 5x x  + x x  - 14x x  - x x  - 8x x  - 8x x  + 2x x  + 4x x  + 2x x  + 4x x  + 3x x  - 7x x  + 2x x  - 3x x })
                             0 3    2 4    3 4      0 5     2 5    3 5    4 5      5     0 6     2 6    4 6      5 6     6    4 7     5 7     6 7     4 8      5 8     6 8     1 2     1 5     0 6     1 6    4 6    5 6     0 7    1 7     2 7     5 7     6 7     1 8     7 8       0     0 2      0 4     2 4    4      0 5    2 5     4 5      0 6     4 6    5 6    0 7     2 7    4 7     5 7     6 7     0 8     4 8     7 8   2 4    3 4    4     2 5     4 5      5 6     6     3 7     4 7      5 7     6 7    3 8    4 8      5 8    6 8      0 4    2 4     2 5     4 5     0 6     2 6    4 6    5 6    4 7     5 7     6 7    4 8     5 8     6 8   0 4    4     1 5     4 5    0 6    1 6     4 6    5 6    4 7     5 7     6 7    4 8     5 8     6 8   2 3    4     4 5    4 6     5 6     6    3 7    4 7     5 7     6 7    4 8     5 8     6 8   1 3     1 5    1 6    4 6     5 6     6    3 7      0 3    3 4    4     0 5     4 5    0 6    4 6     5 6     6    3 7    4 7     5 7     6 7    4 8     5 8     6 8      0 2     2    2 4    4    2 5     4 5      0 6     5 6     2 7     4 7      5 7     6 7     0 8     2 8     4 8     5 8     6 8     7 8      0 1    1 2    1 4     0 6    1 6    4 6    0 7   0 2    1 2     0 4    1 4      1 5    2 5     4 5     0 6     1 6     4 6     2 7     0 8     1 8     5 8     6 8     7 8

o4 : RingMap ringP8 <--- ringP11
i5 : time kernelComponent(phi,1)
     -- used 0.0796423 seconds

o5 = ideal ()

o5 : Ideal of ringP11
i6 : time kernelComponent(phi,2)
     -- used 5.01424 seconds

                           2                                                
o6 = ideal (y y  + y y  + y  + 5y y  + y y  + 5y y  - y y  - 4y y  - 5y y  -
             2 4    3 4    4     2 5    3 5     4 5    1 6     2 6     5 6  
     ------------------------------------------------------------------------
                                                                           
     4y y  - 2y y  - y y  + 4y y  - 5y y  - 4y y  + 3y y  - 4y y  - y y   -
       2 7     4 7    1 8     4 8     5 8     5 9     7 9     8 9    3 10  
     ------------------------------------------------------------------------
                                                                        2
     3y y   - 5y y   - y y   + 4y y   + 5y y  , 3y y  - y y  - 3y y  - y  +
       6 10     8 10    4 11     6 11     8 11    1 3    2 3     3 4    4  
     ------------------------------------------------------------------------
     2y y  - y y  + y y  + 2y y  + 3y y  - 7y y  - 4y y  + 7y y  - 2y y  +
       0 5    3 5    1 6     2 6     5 6     2 7     4 7     1 8     4 8  
     ------------------------------------------------------------------------
     y y  - y y  + 2y y  + 2y y  + y y  - 7y y   + 5y y   - 3y y   - y y   -
      0 9    4 9     5 9     7 9    8 9     0 10     3 10     6 10    0 11  
     ------------------------------------------------------------------------
     2y y   - 2y y  , 7y y  + y y  + 7y y  - y y  + 8y y  - y y  - y y  +
       3 11     4 11    0 1    0 4     1 4    3 4     0 5    3 5    1 6  
     ------------------------------------------------------------------------
     7y y  + 8y y  + y y  + 8y y  - y y  - 8y y  + 7y y  - 8y y  + 7y y  +
       2 6     5 6    2 7     4 7    1 8     4 8     5 9     7 9     8 9  
     ------------------------------------------------------------------------
     y y   - y y   + 8y y   - 7y y   - 7y y   - 7y y  )
      0 10    3 10     6 10     0 11     4 11     6 11

o6 : Ideal of ringP11

An obvious change to the idea of the algorithm allows to perform this computation even when the source of the rational map Φ is a hypersurface of degree d times the degree of the forms defining the map.

i7 : -- phi':phi^(-1)(P^10)--->P^11, restriction of phi:P^8--->P^11 
     -- to the inverse image of a general hyperplane H in P^11
     H=trim ideal random(1,ringP11)

o7 = ideal(810y  + 90y  + 405y  + 90y  + 180y  + 135y  + 270y  + 135y  +
               0      1       2      3       4       5       6       7  
     ------------------------------------------------------------------------
     315y  + 140y  + 126y   + 90y  )
         8       9       10      11

o7 : Ideal of ringP11
i8 : ringHypersurface=ringP8/phi(H); phi'=map(ringHypersurface,ringP8) * phi;

o9 : RingMap ringHypersurface <--- ringP11
i10 : time kernelComponent(phi',1)
     -- used 0.126841 seconds

o10 = ideal(810y  + 90y  + 405y  + 90y  + 180y  + 135y  + 270y  + 135y  +
                0      1       2      3       4       5       6       7  
      -----------------------------------------------------------------------
      315y  + 140y  + 126y   + 90y  )
          8       9       10      11

o10 : Ideal of ringP11

See also

Ways to use kernelComponent :