map(CoincidentRootLocus) -- the map associated to a coincident root locus

Synopsis

Function: map
Usage:

map X
Inputs:
- X, a coincident root locus
Optional inputs:
- Degree => ..., default value null,
- DegreeLift => ..., default value null,
- DegreeMap => ..., default value null,
Outputs:
- a rational map, a unirational parameterization of X

Description

If $X\subset\mathbb{P}^n$ is the coincident root locus associated with the partition $(\lambda_1,\ldots,\lambda_d)$, then we have a map $\mathbb{P}^1\times\cdots\times\mathbb{P}^1\to\mathbb{P}^n$ which sends a $d$-tuple of linear forms $(L_1,\ldots,L_d)$ to ${L_1}^{\lambda_1}\cdots{L_d}^{\lambda_d}$. It is this map that is returned.

i1 : X = coincidentRootLocus {3,2,2}

o1 = CRL(3,2,2)

o1 : Coincident root locus

i2 : f = map X

o2 = -- rational map --
     source: Proj(QQ[t0 , t0 ]) x Proj(QQ[t1 , t1 ]) x Proj(QQ[t2 , t2 ])
                       0    1               0    1               0    1
     target: Proj(QQ[t , t , t , t , t , t , t , t ])
                      0   1   2   3   4   5   6   7
     defining forms: {
                        3  2  2
                      t0 t1 t2 ,
                        0  0  0
                      
                      2  3  2         2  3        2   3  2     2  2
                      -t0 t1 t2 t2  + -t0 t1 t1 t2  + -t0 t0 t1 t2 ,
                      7  0  0  0  1   7  0  0  1  0   7  0  1  0  0
                      
                       1  3  2  2    4  3                1  3  2  2   2  2     2         2  2           2   1     2  2  2
                      --t0 t1 t2  + --t0 t1 t1 t2 t2  + --t0 t1 t2  + -t0 t0 t1 t2 t2  + -t0 t0 t1 t1 t2  + -t0 t0 t1 t2 ,
                      21  0  0  1   21  0  0  1  0  1   21  0  1  0   7  0  1  0  0  1   7  0  1  0  1  0   7  0  1  0  0
                      
                       2  3        2    2  3  2          3  2     2  2   12  2                   3  2     2  2    6     2  2          6     2        2    1  3  2  2
                      --t0 t1 t1 t2  + --t0 t1 t2 t2  + --t0 t0 t1 t2  + --t0 t0 t1 t1 t2 t2  + --t0 t0 t1 t2  + --t0 t0 t1 t2 t2  + --t0 t0 t1 t1 t2  + --t0 t1 t2 ,
                      35  0  0  1  1   35  0  1  0  1   35  0  1  0  1   35  0  1  0  1  0  1   35  0  1  1  0   35  0  1  0  0  1   35  0  1  0  1  0   35  1  0  0
                      
                       1  3  2  2    6  2           2    6  2     2          3     2  2  2   12     2                3     2  2  2    2  3  2          2  3        2
                      --t0 t1 t2  + --t0 t0 t1 t1 t2  + --t0 t0 t1 t2 t2  + --t0 t0 t1 t2  + --t0 t0 t1 t1 t2 t2  + --t0 t0 t1 t2  + --t0 t1 t2 t2  + --t0 t1 t1 t2 ,
                      35  0  1  1   35  0  1  0  1  1   35  0  1  1  0  1   35  0  1  0  1   35  0  1  0  1  0  1   35  0  1  1  0   35  1  0  0  1   35  1  0  1  0
                      
                      1  2     2  2   2     2        2   2     2  2          1  3  2  2    4  3                1  3  2  2
                      -t0 t0 t1 t2  + -t0 t0 t1 t1 t2  + -t0 t0 t1 t2 t2  + --t0 t1 t2  + --t0 t1 t1 t2 t2  + --t0 t1 t2 ,
                      7  0  1  1  1   7  0  1  0  1  1   7  0  1  1  0  1   21  1  0  1   21  1  0  1  0  1   21  1  1  0
                      
                      3     2  2  2   2  3        2   2  3  2
                      -t0 t0 t1 t2  + -t0 t1 t1 t2  + -t0 t1 t2 t2 ,
                      7  0  1  1  1   7  1  0  1  1   7  1  1  0  1
                      
                        3  2  2
                      t0 t1 t2
                        1  1  1
                     }

o2 : MultihomogeneousRationalMap (rational map from PP^1 x PP^1 x PP^1 to PP^7)

i3 : describe f

o3 = rational map defined by multiforms of degree {3, 2, 2}
     source variety: PP^1 x PP^1 x PP^1
     target variety: PP^7
     image: 3-dimensional variety of degree 36 in PP^7 cut out by 364 hypersurfaces of degree 6
     dominance: false
     birationality: false
     coefficient ring: QQ

Ways to use this method: