chowEquations -- Chow equations of a projective variety

Synopsis

Usage:

chowEquations W
Inputs:
- W, a ring element, the Chow form of an irreducible projective variety $X\subset\mathbb{P}^n$
Optional inputs:
- Variable => ..., default value null, specify a name for a variable
Outputs:
- the ideal generated by the Chow equations of $X$

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 0.0476687 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 0.030514 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 0.0645477 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 0.0521498 seconds

o11 = true

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

Ways to use chowEquations :

chowEquations(RingElement)

For the programmer

The object chowEquations is a method function with options.