conormal -- Computes the conormal variety

Synopsis

Usage:

conormal(I)
Inputs:
- I, an ideal, in a polynomial ring $k[x_1,\dots, x_n]$ for a field $k$, defining a closed variety in $k^n$ (or in $\PP^n$ if the ideal is homogeneous).
Outputs:
- C, an ideal defining the conormal variety in $k^n \times \PP^{n-1}$.

Description

For a complex projective variety $X=V(I)\subset \PP^n$ this command computes the ideal of the conormal variety $Con(X)$ in $k^n \times \PP^{n-1}$.

i1 : S=QQ[x..z]

o1 = S

o1 : PolynomialRing

i2 : I=ideal(y^2*z-x^2)

            2     2
o2 = ideal(y z - x )

o2 : Ideal of S

i3 : conormal I

                                           2    2     2                  
o3 = ideal (y*v  - 2z*v , x*v  + 2z*v , z*v  - v , y*v  - 2v v , y*z*v  +
               1       2     0       2     0    1     0     1 2       0  
     ------------------------------------------------------------------------
            2             2     2
     x*v , y v  + 2x*v , y z - x )
        1     0       2

o3 : Ideal of QQ[x..z, v ..v ]
                        0   2

For the programmer

The object conormal is a method function.