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
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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
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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
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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
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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
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The object conormalVariety is a method function with options.