segre -- Segre embedding

Synopsis

Usage:

segre phi

segre R
Inputs:
- phi, a rational map, with source a closed subvariety $X\subseteq\mathbb{P}^{n_1}\times\mathbb{P}^{n_2}\times\cdots\times\mathbb{P}^{n_k}$ of a product of projective spaces
- R, a quotient ring, or a polynomial ring, the coordinate ring of the subvariety $X\subseteq\mathbb{P}^{n_1}\times\mathbb{P}^{n_2}\times\cdots\times\mathbb{P}^{n_k}$
Outputs:
- a rational map, the restriction to $X$ of the Segre embedding of $\mathbb{P}^{n_1}\times\mathbb{P}^{n_2}\times\cdots\times\mathbb{P}^{n_k}$, where the linear span of the image is identified with a projective space

Description

More properly, this method accepts and returns objects of the class MultihomogeneousRationalMap.

i1 : phi = first graph quadroQuadricCremonaTransformation(3,1)

o1 = -- rational map --
     source: subvariety of Proj(QQ[x , x , x , x ]) x Proj(QQ[y , y , y , y ]) defined by
                                    0   1   2   3              0   1   2   3
             {
              x y  - x y ,
               3 2    2 3
              
              x y  + x y ,
               3 1    1 3
              
              x y  + x y ,
               2 1    1 2
              
              x y  - x y  + x y  + x y
               0 0    1 1    2 2    3 3
             }
     target: Proj(QQ[x, y, z, t])
     defining forms: {
                      x ,
                       0
                      
                      x ,
                       1
                      
                      x ,
                       2
                      
                      x
                       3
                     }

o1 : MultihomogeneousRationalMap (birational map from threefold in PP^3 x PP^3 to PP^3)

i2 : segre phi

o2 = -- rational map --
     source: subvariety of Proj(QQ[x , x , x , x ]) x Proj(QQ[y , y , y , y ]) defined by
                                    0   1   2   3              0   1   2   3
             {
              x y  - x y ,
               3 2    2 3
              
              x y  + x y ,
               3 1    1 3
              
              x y  + x y ,
               2 1    1 2
              
              x y  - x y  + x y  + x y
               0 0    1 1    2 2    3 3
             }
     target: Proj(QQ[t , t , t , t , t , t , t , t , t , t , t  , t  ])
                      0   1   2   3   4   5   6   7   8   9   10   11
     defining forms: {
                      x y ,
                       0 1
                      
                      x y ,
                       0 2
                      
                      x y ,
                       0 3
                      
                      x y ,
                       1 0
                      
                      x y ,
                       1 1
                      
                      x y ,
                       2 0
                      
                      -x y ,
                        1 2
                      
                      x y ,
                       2 2
                      
                      x y ,
                       3 0
                      
                      -x y ,
                        1 3
                      
                      x y ,
                       2 3
                      
                      x y
                       3 3
                     }

o2 : MultihomogeneousRationalMap (rational map from threefold in PP^3 x PP^3 to PP^11)

Ways to use segre:

segre(PolynomialRing)
segre(QuotientRing)
segre(RationalMap)

For the programmer

The object segre is a method function.