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 0.0913076 seconds 2 2 2 2 2 2 4 2 2 2 2 2 3 2 2 3 3 2 o3 = ideal (x x + x x + x x + x , x x x x + x x x + x x , x x x + x x x + x x - 2x x x - 2x x x - x x , x x + 2x x x 0 3 1 3 2 3 3 0 1 2 3 1 2 3 2 3 0 2 3 1 2 3 2 3 0 1 3 1 2 3 2 3 1 3 1 2 3 ---------------------------------------------------------------------------------------------------------------------------- 2 2 2 3 2 2 2 2 3 2 2 2 2 2 3 2 2 - x x x + x x + x x , x x x + x x x + 2x x x + 3x x x + 2x x , x x x - x x x + x x x - x x , x x - x x x + x x x 0 2 3 2 3 1 3 0 1 3 1 2 3 0 1 3 1 2 3 2 3 0 1 3 1 2 3 0 2 3 2 3 0 3 1 2 3 0 2 3 ---------------------------------------------------------------------------------------------------------------------------- 2 2 3 3 2 3 3 2 2 3 2 2 2 2 4 2 2 3 3 + 4x x x + 3x x x + x x + 4x 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 + 2x x - 0 1 3 1 2 3 0 3 2 3 0 1 2 1 2 2 3 0 1 3 1 2 3 2 3 0 2 1 2 2 2 3 1 2 1 2 ---------------------------------------------------------------------------------------------------------------------------- 2 3 2 2 3 2 2 2 2 2 3 2 3 2 2 3 x x x + x x + 3x x x + 4x x x + 3x 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 + x x , 0 2 3 2 3 0 1 3 1 2 3 2 3 0 1 2 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 2 2 3 3 2 2 3 2 x x - x x + x x - x x x + x x x , x + 2x x + 2x x x + x x + x x , x x - 2x x + x x x + x x x - x x - 2x x x - 0 2 1 2 0 2 1 2 3 0 2 3 1 1 2 1 2 3 1 3 2 3 0 1 1 2 1 2 3 0 2 3 2 3 0 1 3 ---------------------------------------------------------------------------------------------------------------------------- 2 3 2 2 2 2 2 2 2 3 3 2 2 3 4 4 2 2 2 2 2 4 3x x x - 2x x , x x - x x - 2x x x - x x , x x + x x - x x x - x x x - x x , x - x + 2x x x - x x - x x - x ) 1 2 3 2 3 0 1 1 2 1 2 3 2 3 0 1 1 2 1 2 3 0 2 3 2 3 0 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 0.0565409 seconds 4 3 2 3 2 2 3 2 2 2 2 2 2 2 3 2 3 3 2 4 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 , x x x x - x x , x x x - x x , x x - 4x 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 0 1 2 3 0 3 1 2 3 0 3 1 3 0 1 2 3 ---------------------------------------------------------------------------------------------------------------------------- 2 2 3 2 2 2 3 5 3 2 4 2 2 4 2 2 3 2 3 3 3x 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 - x x x x , x x - x x x x , x x x - x x x , 0 2 3 0 1 3 0 1 2 3 0 1 3 0 2 3 2 0 3 1 2 0 2 3 0 2 0 1 2 3 1 2 0 1 2 3 0 1 2 0 2 3 ---------------------------------------------------------------------------------------------------------------------------- 2 3 3 3 2 3 2 2 4 4 3 3 4 2 2 3 2 5 4 4 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 - x x , x x x - x x , x - x x , x x - x x , x x - 0 2 0 1 3 1 2 0 2 3 0 1 2 0 3 1 2 0 1 3 0 1 2 0 3 0 1 2 0 2 1 0 3 0 1 0 2 0 1 ---------------------------------------------------------------------------------------------------------------------------- 3 3 2 4 x x x , x x - x x ) 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 0.11965 seconds 2 2 o10 = (x x + x x , x - 4x x + 2x x + x , 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 0,2,3 1,2,3 o10 : Sequence |
i11 : time (Q,Q,Q) == (chowEquations(W0,0),chowEquations(W1,1),chowEquations(W2,2)) -- used 0.103089 seconds o11 = true |
Note that chowEquations(W,0) is not the same as chowEquations W.
The object chowEquations is a method function with options.