# tangentialChowForm -- higher Chow forms of a projective variety

## Synopsis

• Usage:
tangentialChowForm(I,s)
• Inputs:
• I, an ideal, a homogeneous ideal defining a projective variety $X=V(I)\subset\mathbb{P}^n$, say of dimension $k$
• s, an integer
• Optional inputs:
• AffineChartGrass => ..., default value true, use an affine chart on the Grassmannian
• AffineChartProj => ..., default value true, use an affine chart on the projective space
• AssumeOrdinary => ..., default value null, whether the expected codimension is 1
• Duality => ..., default value null, whether to use dual Plücker coordinates
• SingularLocus => ..., default value null, pass the singular locus of the variety
• Variable => ..., default value null, specify a name for a variable
• Outputs:
• , or an ideal (if there is more than one generator) in the coordinate ring of the Grassmannian $\mathbb{G}(n-k-1+s,\mathbb{P}^n)$ in the Plücker embedding, representing the higher associated subvariety $Z_s(X)$

## Description

For a projective variety $X\subset\mathbb{P}^n$ of dimension $k$, the $s$-th associated subvariety $Z_s(X)\subset\mathbb{G}(n-k-1+s,\mathbb{P}^n)$ (also called tangential Chow form) is defined to be the closure of the set of $(n-k-1+s)$-dimensional subspaces $L\subset \mathbb{P}^n$ such that $L\cap X\neq\emptyset$ and $dim(L\cap T_x(X))\geq s$ for some smooth point $x\in L\cap X$, where $T_x(X)$ denotes the embedded tangent space to $X$ at $x$. In particular, $Z_0(X)\subset\mathbb{G}(n-k-1,\mathbb{P}^n)$ is defined by the Chow form of $X$, while $Z_k(X)\subset\mathbb{G}(n-1,\mathbb{P}^n)$ is identified to the dual variety $X^{*}\subset{\mathbb{P}^n}^{*}=\mathbb{G}(0,{\mathbb{P}^n}^{*})$ via the duality of Grassmannians $\mathbb{G}(0,{\mathbb{P}^n}^{*})=\mathbb{G}(n-1,\mathbb{P}^n)$. For details we refer to the third chapter of Discriminants, Resultants, and Multidimensional Determinants, by Israel M. Gelfand, Mikhail M. Kapranov and Andrei V. Zelevinsky.

The algorithm used are standard, based on projections of suitable incidence varieties. Here are some of the options available that could speed up the computation.

Duality Taking into account the duality of Grassmannians, one can perform the computation in $\mathbb{G}(k-s,n)$ and then passing to $\mathbb{G}(n-k-1+s,n)$. This is done by default when it seems advantageous.

AffineChartGrass If one of the standard coordinate charts on the Grassmannian is specified, then the internal computation is done on that chart. By default, a random chart is used. Set this to false to not use any chart.

AffineChartProj This is quite similar to AffineChartGrass, but it allows to specify one of the standard coordinate charts on the projective space. You should set this to false for working with reducible or degenerate varieties.

AssumeOrdinary Set this to true if you know that $Z_s(X)$ is a hypersurface (by default is already true if $s=0$).

 i1 : -- cubic rational normal scroll surface in P^4=G(0,4) use Grass(0,4,Variable=>p); S = minors(2,matrix{{p_0,p_2,p_3},{p_1,p_3,p_4}}) 2 o2 = ideal (- p p + p p , - p p + p p , - p + p p ) 1 2 0 3 1 3 0 4 3 2 4 o2 : Ideal of QQ[p ..p ] 0 4 i3 : -- 0-th associated hypersurface of S in G(1,4) (Chow form) time tangentialChowForm(S,0) -- used 0.11833 seconds 2 2 o3 = p p - p p p - p p p + p p p + p p + 1,3 2,3 1,2 1,3 2,4 0,3 1,3 2,4 0,2 1,4 2,4 1,2 3,4 ------------------------------------------------------------------------ 2 p p - 2p p p - p p p 0,3 3,4 0,1 2,3 3,4 0,2 0,4 3,4 QQ[p ..p , p , p , p , p , p , p , p , p ] 0,1 0,2 1,2 0,3 1,3 2,3 0,4 1,4 2,4 3,4 o3 : ---------------------------------------------------------------------------------------------------------------------------------------------------------------- (p p - p p + p p , p p - p p + p p , p p - p p + p p , p p - p p + p p , p p - p p + p p ) 2,3 1,4 1,3 2,4 1,2 3,4 2,3 0,4 0,3 2,4 0,2 3,4 1,3 0,4 0,3 1,4 0,1 3,4 1,2 0,4 0,2 1,4 0,1 2,4 1,2 0,3 0,2 1,3 0,1 2,3 i4 : -- 1-th associated hypersurface of S in G(2,4) time tangentialChowForm(S,1) -- used 0.302977 seconds 2 2 2 2 3 2 2 o4 = p p + p p - 2p p + p p - 1,2,3 1,2,4 0,2,4 1,2,4 0,2,3 1,2,4 0,2,4 0,3,4 ------------------------------------------------------------------------ 3 3 3 4p p - 4p p - 2p p + 0,2,3 0,3,4 1,2,3 1,3,4 0,2,4 1,3,4 ------------------------------------------------------------------------ 8p p p p - 2p p p p + 0,2,3 1,2,3 1,2,4 1,3,4 0,2,3 0,2,4 1,2,4 1,3,4 ------------------------------------------------------------------------ 2 2 2 8p p p p - 8p p - 2p p p - 0,2,3 0,2,4 0,3,4 1,3,4 0,2,3 1,3,4 0,1,4 0,2,4 2,3,4 ------------------------------------------------------------------------ 2p p p p + 8p p p p - 0,1,3 1,2,3 1,2,4 2,3,4 0,1,3 0,2,4 1,2,4 2,3,4 ------------------------------------------------------------------------ 2 2 2p p p + 10p p p p - 12p p p 0,1,2 1,2,4 2,3,4 0,1,3 0,2,4 0,3,4 2,3,4 0,1,2 0,3,4 2,3,4 ------------------------------------------------------------------------ 2 2 - 20p p p p + 12p p p p + p p 0,1,3 0,2,3 1,3,4 2,3,4 0,1,2 1,2,3 1,3,4 2,3,4 0,1,3 2,3,4 ------------------------------------------------------------------------ 2 + 12p p p 0,1,2 0,1,4 2,3,4 QQ[p ..p , p , p , p , p , p , p , p , p ] 0,1,2 0,1,3 0,2,3 1,2,3 0,1,4 0,2,4 1,2,4 0,3,4 1,3,4 2,3,4 o4 : ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- (p p - p p + p p , p p - p p + p p , p p - p p + p p , p p - p p + p p , p p - p p + p p ) 1,2,4 0,3,4 0,2,4 1,3,4 0,1,4 2,3,4 1,2,3 0,3,4 0,2,3 1,3,4 0,1,3 2,3,4 1,2,3 0,2,4 0,2,3 1,2,4 0,1,2 2,3,4 1,2,3 0,1,4 0,1,3 1,2,4 0,1,2 1,3,4 0,2,3 0,1,4 0,1,3 0,2,4 0,1,2 0,3,4 i5 : -- 2-th associated hypersurface of S in G(3,4) (parameterizing tangent hyperplanes to S) time tangentialChowForm(S,2) -- used 0.114329 seconds 2 2 o5 = p p - p p p + p p 0,1,3,4 0,2,3,4 0,1,2,4 0,2,3,4 1,2,3,4 0,1,2,3 1,2,3,4 o5 : QQ[p ..p , p , p , p ] 0,1,2,3 0,1,2,4 0,1,3,4 0,2,3,4 1,2,3,4 i6 : -- we get the dual hypersurface of S in G(0,4) by dualizing time S' = ideal dualize tangentialChowForm(S,2) -- used 0.126361 seconds 2 2 o6 = ideal(p p - p p p + p p ) 1 2 0 1 3 0 4 o6 : Ideal of QQ[p ..p ] 0 4 i7 : -- we then can recover S time assert(dualize tangentialChowForm(S',3) == S) -- used 0.302813 seconds