# chowEquations -- Chow equations of a projective variety

## Synopsis

• Usage:
chowEquations W
• Inputs:
• W, , 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 0.182921 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 0.190644 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 0.323569 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 0.2043 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 .