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Resultants :: tangentialChowForm

tangentialChowForm -- higher Chow forms of a projective variety

Synopsis

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 0.042828 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 0.0550171 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 0.0246971 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 0.0504611 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 0.056514 seconds

See also

Ways to use tangentialChowForm :

For the programmer

The object tangentialChowForm is a method function with options.