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Resultants :: tangentialChowForm

tangentialChowForm -- higher Chow forms of a projective variety

Synopsis

Description

For a projective variety X⊂ℙn of dimension k, the s-th associated subvariety Zs(X)⊂G(n-k-1+s,ℙn) (also called tangential Chow form) is defined to be the closure of the set of (n-k-1+s)-dimensional subspaces L⊂ℙn such that L∩X≠Ø and dim(L∩Tx(X))≥s for some smooth point x∈L∩X, where Tx(X) denotes the embedded tangent space to X at x. In particular, Z0(X)⊂G(n-k-1,ℙn) is defined by the Chow form of X, while Zk(X)⊂G(n-1,ℙn) is identified to the dual variety X*⊂ℙn*=G(0,ℙn*) via the duality of Grassmannians G(0,ℙn*)=G(n-1,ℙn). 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 G(k-s,n) and then passing to 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 Zs(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 , p , p , p ]
                  0   1   2   3   4
i3 : -- 0-th associated hypersurface of S in G(1,4) (Chow form)
     time tangentialChowForm(S,0)
     -- used 0.0217617 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 0.0927697 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 0.0231241 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 0.0270482 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 , p , p , p ]
                  0   1   2   3   4
i7 : -- we then can recover S
     time dualize tangentialChowForm(S',3) == S
     -- used 0.0407136 seconds

o7 = true

See also

Ways to use tangentialChowForm :