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Resultants :: cayleyTrick

cayleyTrick -- Cayley trick

Synopsis

Description

Let $X\subset\mathbb{P}^n$ be a $k$-dimensional projective variety. Consider the product $W = X\times\mathbb{P}^k$ as a subvariety of $\mathbb{P}(Mat(k+1,n+1))$, the projectivization of the space of $(k+1)\times (n+1)$-matrices, and consider the projection $p:\mathbb{P}(Mat(k+1,n+1))\dashrightarrow\mathbb{G}(k,n)=\mathbb{G}(n-k-1,n)$. Then the "Cayley trick" states that the dual variety $W^*$ of $W$ equals the closure of $p^{-1}(Z_0(X))$, where $Z_0(X)\subset\mathbb{G}(n-k-1,n)$ is the Chow hypersurface of $X$. The defining form of $W^*$ is also called the $X$-resultant. For details and proof, see Multiplicative properties of projectively dual varieties, by J. Weyman and A. Zelevinsky; see also Coisotropic hypersurfaces in Grassmannians, by K. Kohn.

In the example below, we apply the method to the quadric $\mathbb{P}^1\times\mathbb{P}^1\subset\mathbb{P}^3$.

i1 : QQ[x_0..x_3]; P1xP1 = ideal(x_0*x_1-x_2*x_3)

o2 = ideal(x x  - x x )
            0 1    2 3

o2 : Ideal of QQ[x ..x ]
                  0   3
i3 : time (P1xP1xP2,P1xP1xP2') = cayleyTrick(P1xP1,2);
     -- used 0.0586885 seconds

In the next example, we calculate the defining ideal of $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1\subset\mathbb{P}^7$ and that of its dual variety.

i4 : time (P1xP1xP1,P1xP1xP1') = cayleyTrick(P1xP1,1)
     -- used 0.0439699 seconds

                                                                           
o4 = (ideal (x   x    - x   x   , x   x    - x   x   , x   x    - x   x   ,
              0,3 1,2    0,2 1,3   1,0 1,1    1,2 1,3   0,3 1,1    0,1 1,3 
     ------------------------------------------------------------------------
                                                                            
     x   x    - x   x   , x   x    - x   x   , x   x    - x   x   , x   x   
      0,2 1,1    0,1 1,2   0,0 1,1    0,2 1,3   0,3 1,0    0,0 1,3   0,2 1,0
     ------------------------------------------------------------------------
                                                                   2   2    
     - x   x   , x   x    - x   x   , x   x    - x   x   ), ideal(x   x    -
        0,0 1,2   0,1 1,0    0,2 1,3   0,0 0,1    0,2 0,3          0,1 1,0  
     ------------------------------------------------------------------------
                                              2   2                        
     2x   x   x   x    + 4x   x   x   x    + x   x    - 2x   x   x   x    -
       0,0 0,1 1,0 1,1     0,2 0,3 1,0 1,1    0,0 1,1     0,1 0,3 1,0 1,2  
     ------------------------------------------------------------------------
                          2   2                                            
     2x   x   x   x    + x   x    - 2x   x   x   x    - 2x   x   x   x    +
       0,0 0,3 1,1 1,2    0,3 1,2     0,1 0,2 1,0 1,3     0,0 0,2 1,1 1,3  
     ------------------------------------------------------------------------
                                              2   2
     4x   x   x   x    - 2x   x   x   x    + x   x   ))
       0,0 0,1 1,2 1,3     0,2 0,3 1,2 1,3    0,2 1,3

o4 : Sequence

If the option Duality is set to true, then the method applies the so-called "dual Cayley trick".

i5 : time cayleyTrick(P1xP1,1,Duality=>true);
     -- used 0.0840145 seconds
i6 : assert(oo == (P1xP1xP1,P1xP1xP1'))
i7 : time cayleyTrick(P1xP1,2,Duality=>true);
     -- used 0.0758903 seconds
i8 : assert(oo == (P1xP1xP2,P1xP1xP2'))

See also

Ways to use cayleyTrick :

For the programmer

The object cayleyTrick is a method function with options.