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# conormalVariety -- conormal variety

## Synopsis

• Usage:
conormalVariety I
• Inputs:
• I, an ideal, a homogeneous ideal defining a projective variety $X=V(I)\subset\mathbb{P}^n$
• Optional inputs:
• SingularLocus => ..., default value null, pass the singular locus of the variety
• Strategy => ..., default value "Saturate",
• Variable => ..., default value null, specify a name for a variable
• Outputs:
• an ideal, the bihomogeneous ideal of the conormal variety $Con(X)\subset\mathbb{P}^n\times{\mathbb{P}^n}^{*}$ of $X$

## Description

The conormal variety $Con(X)$ of a projective variety $X\subset\mathbb{P}^n$ is the Zariski closure in $\mathbb{P}^n\times{\mathbb{P}^n}^{*}$ of the set of tuples $(x,H)$ where $x$ is a regular point of $X$ and $H$ is a hyperplane in $\mathbb{P}^n$ containing the embedded tangent space to $X$ at $x$. The dual variety of $X$ is the image of $Con(X)\subset\mathbb{P}^n\times{\mathbb{P}^n}^{*}$ under projection onto the second factor ${\mathbb{P}^n}^{*}$.

 i1 : X = kernel veronese(1,3) 2 2 o1 = ideal (x - x x , x x - x x , x - x x ) 2 1 3 1 2 0 3 1 0 2 o1 : Ideal of QQ[x ..x ] 0 3 i2 : conormalVariety X o2 = ideal (x x + 2x x + 3x x , x x + 2x x + 0,1 1,1 0,2 1,2 0,3 1,3 0,0 1,1 0,1 1,2 ------------------------------------------------------------------------ 3x x , 3x x + 2x x + x x , x x - x x - 0,2 1,3 0,1 1,0 0,2 1,1 0,3 1,2 0,0 1,0 0,2 1,2 ------------------------------------------------------------------------ 2 2 2x x , x - x x , x x - x x , x - x x , 0,3 1,3 0,2 0,1 0,3 0,1 0,2 0,0 0,3 0,1 0,0 0,2 ------------------------------------------------------------------------ 2 2 3 3 2 2 x x - 4x x - 4x x + 18x x x x - 27x x ) 1,1 1,2 1,0 1,2 1,1 1,3 1,0 1,1 1,2 1,3 1,0 1,3 o2 : Ideal of QQ[x ..x ] 0,0 1,3