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Cremona :: RationalMap ^** Ideal

RationalMap ^** Ideal -- inverse image via a rational map

Synopsis

Description

In most cases this is equivalent to phi^* I, which is faster but may not take into account other representations of the map.

In the example below, we apply the method to check the birationality of a map (deterministically).

i1 : phi = quadroQuadricCremonaTransformation(5,1)

o1 = -- rational map --
     source: Proj(QQ[x, y, z, t, u, v])
     target: Proj(QQ[x, y, z, t, u, v])
     defining forms: {
                             2
                      y*z - v ,
                      
                             2
                      x*z - u ,
                      
                             2
                      x*y - t ,
                      
                      - z*t + u*v,
                      
                      - y*u + t*v,
                      
                      t*u - x*v
                     }

o1 : RationalMap (Cremona transformation of PP^5 of type (2,2))
i2 : K := frac(QQ[vars(0..5)]); phi = phi ** K

o3 = -- rational map --
     source: Proj(frac(QQ[a..f])[x, y, z, t, u, v])
     target: Proj(frac(QQ[a..f])[x, y, z, t, u, v])
     defining forms: {
                             2
                      y*z - v ,
                      
                             2
                      x*z - u ,
                      
                             2
                      x*y - t ,
                      
                      - z*t + u*v,
                      
                      - y*u + t*v,
                      
                      t*u - x*v
                     }

o3 : RationalMap (quadratic rational map from PP^5 to PP^5)
i4 : p = trim minors(2,(vars K)||(vars source phi))

                -e       -d       -c       -b       -a
o4 = ideal (u + --v, t + --v, z + --v, y + --v, x + --v)
                 f        f        f        f        f

o4 : Ideal of frac(QQ[a..f])[x, y, z, t, u, v]
i5 : q = phi p

                                                         2                2  
                b*e - d*f       c*d - e*f       - a*b + d        - a*c + e   
o5 = ideal (u + ---------v, t + ---------v, z + ----------v, y + ----------v,
                d*e - a*f       d*e - a*f        d*e - a*f        d*e - a*f  
     ------------------------------------------------------------------------
                  2
         - b*c + f
     x + ----------v)
          d*e - a*f

o5 : Ideal of frac(QQ[a..f])[x, y, z, t, u, v]
i6 : time phi^** q
     -- used 0.34991 seconds

                -e       -d       -c       -b       -a
o6 = ideal (u + --v, t + --v, z + --v, y + --v, x + --v)
                 f        f        f        f        f

o6 : Ideal of frac(QQ[a..f])[x, y, z, t, u, v]
i7 : oo == p

o7 = true

See also