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conormalVariety -- conormal variety

Synopsis

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

See also

Ways to use conormalVariety :

For the programmer

The object conormalVariety is a method function with options.