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Resultants :: chowEquations

chowEquations -- Chow equations of a projective variety

Synopsis

Description

Given the Chow form $Z_0(X)\subset\mathbb{G}(n-k-1,n)$ of an irreducible projective $k$-dimensional variety $X\subset\mathbb{P}^n$, one can recover a canonical system of equations, called Chow equations, that always define $X$ set-theoretically, and also scheme-theoretically whenever $X$ is smooth. For details, see chapter 3, section 2C of Discriminants, Resultants, and Multidimensional Determinants, by Israel M. Gelfand, Mikhail M. Kapranov and Andrei V. Zelevinsky.

i1 : P3 = Grass(0,3,ZZ/11,Variable=>x);
i2 : -- an elliptic quartic curve
     C = ideal(x_0^2+x_1^2+x_2^2+x_3^2,x_0*x_1+x_1*x_2+x_2*x_3)

             2    2    2    2
o2 = ideal (x  + x  + x  + x , x x  + x x  + x x )
             0    1    2    3   0 1    1 2    2 3

o2 : Ideal of P3
i3 : -- Chow equations of C
     time eqsC = chowEquations chowForm C
     -- used 0.0608986 seconds

             2 2    2 2    2 2    4                2      2 2   2      
o3 = ideal (x x  + x x  + x x  + x , x x x x  + x x x  + x x , x x x  +
             0 3    1 3    2 3    3   0 1 2 3    1 2 3    2 3   0 2 3  
     ------------------------------------------------------------------------
      2        3           2         2      3   3         2          2    2 2
     x x x  + x x  - 2x x x  - 2x x x  - x x , x x  + 2x x x  - x x x  + x x 
      1 2 3    2 3     0 1 3     1 2 3    2 3   1 3     1 2 3    0 2 3    2 3
     ------------------------------------------------------------------------
          3     2      2             2         2       3   2          2    
     + x x , x x x  + x x x  + 2x x x  + 3x x x  + 2x x , x x x  - x x x  +
        1 3   0 1 3    1 2 3     0 1 3     1 2 3     2 3   0 1 3    1 2 3  
     ------------------------------------------------------------------------
          2    2 2   3      2          2           2         2      3  
     x x x  - x x , x x  - x x x  + x x x  + 4x x x  + 3x x x  + x x  +
      0 2 3    2 3   0 3    1 2 3    0 2 3     0 1 3     1 2 3    0 3  
     ------------------------------------------------------------------------
         3       2      3    3          2        2      3   2 2    2 2    4  
     4x x , x x x  + x x  + x x  + x x x  + x x x  + x x , x x  + x x  + x  +
       2 3   0 1 2    1 2    2 3    0 1 3    1 2 3    2 3   0 2    1 2    2  
     ------------------------------------------------------------------------
      2 2   3         3      2      3           2         2       3     2    
     x x , x x  + 2x x  - x x x  + x x  + 3x x x  + 4x x x  + 3x x , x x x  +
      2 3   1 2     1 2    0 2 3    2 3     0 1 3     1 2 3     2 3   0 1 2  
     ------------------------------------------------------------------------
      2 2      2     2          3      2      3          2        2      3 
     x x  + x x x , x x x  - x x  + x x x  - x x  + x x x  + x x x  + x x ,
      1 2    1 2 3   0 1 2    1 2    0 2 3    2 3    0 1 3    1 2 3    2 3 
     ------------------------------------------------------------------------
      3      2 2      3      2          2   4     2 2       2      2 2  
     x x  - x x  + x x  - x x x  + x x x , x  + 2x x  + 2x x x  + x x  +
      0 2    1 2    0 2    1 2 3    0 2 3   1     1 2     1 2 3    1 3  
     ------------------------------------------------------------------------
      2 2     3       3    2          2      3           2         2       3 
     x x , x x  - 2x x  + x x x  + x x x  - x x  - 2x x x  - 3x x x  - 2x x ,
      2 3   0 1     1 2    1 2 3    0 2 3    2 3     0 1 3     1 2 3     2 3 
     ------------------------------------------------------------------------
      2 2    2 2       2      2 2   3        3    2          2        3   4  
     x x  - x x  - 2x x x  - x x , x x  + x x  - x x x  - x x x  - x x , x  -
      0 1    1 2     1 2 3    2 3   0 1    1 2    1 2 3    0 2 3    2 3   0  
     ------------------------------------------------------------------------
      4       2      2 2    2 2    4
     x  + 2x x x  - x x  - x x  - x )
      2     1 2 3    1 3    2 3    3

o3 : Ideal of P3
i4 : C == saturate eqsC

o4 = true
i5 : -- a singular irreducible curve 
     D = ideal(x_1^2-x_0*x_2,x_2^3-x_0*x_1*x_3,x_1*x_2^2-x_0^2*x_3)

             2          3              2    2
o5 = ideal (x  - x x , x  - x x x , x x  - x x )
             1    0 2   2    0 1 3   1 2    0 3

o5 : Ideal of P3
i6 : -- Chow equations of D
     time eqsD = chowEquations chowForm D
     -- used 0.0314046 seconds

             4      3 2     3        2 2     3      2   2   2 2      2   2 
o6 = ideal (x x  - x x , x x x  - x x x , x x x  - x x x , x x x  - x x x ,
             2 3    1 3   1 2 3    0 1 3   0 2 3    0 1 3   1 2 3    0 1 3 
     ------------------------------------------------------------------------
          2      3 2   3        3 2   4         2         2 2       3    
     x x x x  - x x , x x x  - x x , x x  - 4x x x x  + 3x x x , x x x  -
      0 1 2 3    0 3   1 2 3    0 3   1 3     0 1 2 3     0 2 3   0 1 3  
     ------------------------------------------------------------------------
      2         2 2      3       5    3 2     4    2 2       4    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 ,
      0 1 2 3   0 1 3    0 2 3   2    0 3   1 2    0 2 3   0 2    0 1 2 3 
     ------------------------------------------------------------------------
      2 3    2             3    3       2 3    3       3 2    3       2   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 x  -
      1 2    0 1 2 3   0 1 2    0 2 3   0 2    0 1 3   1 2    0 2 3   0 1 2  
     ------------------------------------------------------------------------
      4     4      3         3      4     2 2      3 2   5    4       4  
     x x , x x  - x x x , x x x  - x x , x x x  - x x , x  - x x , x x  -
      0 3   1 2    0 1 3   0 1 2    0 3   0 1 2    0 2   1    0 3   0 1  
     ------------------------------------------------------------------------
      3 2   2 3    3       3 2    4
     x x , x x  - x x x , x x  - x x )
      0 2   0 1    0 1 2   0 1    0 2

o6 : Ideal of P3
i7 : D == saturate eqsD

o7 = false
i8 : D == radical eqsD

o8 = true

Actually, one can use chowEquations to recover a variety $X$ from some other of its tangential Chow forms as well. This is based on generalizations of the "Cayley trick", see Multiplicative properties of projectively dual varieties, by J. Weyman and A. Zelevinsky; see also Coisotropic hypersurfaces in Grassmannians, by K. Kohn. For instance,

i9 : Q = ideal(x_0*x_1+x_2*x_3)

o9 = ideal(x x  + x x )
            0 1    2 3

o9 : Ideal of P3
i10 : -- tangential Chow forms of Q
      time (W0,W1,W2) = (tangentialChowForm(Q,0),tangentialChowForm(Q,1),tangentialChowForm(Q,2))
     -- used 0.0767923 seconds

                     2                              2
o10 = (x x  + x x , x    - 4x   x    + 2x   x    + x   , x     x      +
        0 1    2 3   0,1     0,2 1,3     0,1 2,3    2,3   0,1,2 0,1,3  
      -----------------------------------------------------------------------
      x     x     )
       0,2,3 1,2,3

o10 : Sequence
i11 : time (Q,Q,Q) == (chowEquations(W0,0),chowEquations(W1,1),chowEquations(W2,2))
     -- used 0.0514963 seconds

o11 = true

Note that chowEquations(W,0) is not the same as chowEquations W.

Ways to use chowEquations :

For the programmer

The object chowEquations is a method function with options.