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MultiprojectiveVarieties > MultiprojectiveVariety > projectiveVariety

projectiveVariety -- the closed multi-projective subvariety defined by a multi-homogeneous ideal

Synopsis

Description

Equivalently, one can give as input the coordinate ring of the projective variety, that is, the quotient of $R$ by (the multisaturation of) $I$.

In the example, we take a complete intersection $X\subset\mathbb{P}^{2}\times\mathbb{P}^{3}\times\mathbb{P}^{1}$ of two hypersurfaces of multidegrees $(2,1,0)$ and $(1,0,1)$.

i1 : K = ZZ/333331;
i2 : R = K[x_0..x_2,y_0..y_3,z_0,z_1,Degrees=>{3:{1,0,0},4:{0,1,0},2:{0,0,1}}];
i3 : I = ideal(random({2,1,0},R),random({1,0,1},R))

                    2                         2                  
o3 = ideal (- 34043x y  + 74106x x y  + 52821x y  - 47435x x y  +
                    0 0         0 1 0         1 0         0 2 0  
     ------------------------------------------------------------------------
                          2           2                          2    
     123091x x y  - 66080x y  + 91969x y  - 54528x x y  + 106535x y  -
            1 2 0         2 0         0 1         0 1 1          1 1  
     ------------------------------------------------------------------------
                                         2           2                  
     35766x x y  + 120182x x y  + 159079x y  + 69319x y  - 62743x x y  +
           0 2 1          1 2 1          2 1         0 2         0 1 2  
     ------------------------------------------------------------------------
            2                                      2           2    
     136098x y  - 66116x x y  - 96699x x y  + 9398x y  + 92232x y  +
            1 2         0 2 2         1 2 2        2 2         0 3  
     ------------------------------------------------------------------------
                          2                                       2
     54291x x y  + 155574x y  + 45133x x y  - 77273x x y  - 25242x y ,
           0 1 3          1 3         0 2 3         1 2 3         2 3 
     ------------------------------------------------------------------------
     86018x z  - 125857x z  + 130921x z  - 106029x z  + 5398x z  - 35792x z )
           0 0          1 0          2 0          0 1        1 1         2 1

o3 : Ideal of R
i4 : X = projectiveVariety I

o4 = X

o4 : ProjectiveVariety, 4-dimensional subvariety of PP^2 x PP^3 x PP^1
i5 : ? X -- short description

o5 = 4-dimensional subvariety of PP^2 x PP^3 x PP^1 cut out by 2
     hypersurfaces of multi-degrees (1,0,1)^1 (2,1,0)^1
i6 : describe X -- long description

o6 = ambient:.............. PP^2 x PP^3 x PP^1
     dim:.................. 4
     codim:................ 2
     degree:............... 34
     multidegree:.......... 2*T_0^2+T_0*T_1+2*T_0*T_2+T_1*T_2
     generators:........... (1,0,1)^1 (2,1,0)^1 
     purity:............... true
     dim sing. l.:......... -1
     Segre embedding:...... map to PP^19 ⊂ PP^23

Below, we calculate the image of $X$ via the Segre embedding of $\mathbb{P}^{2}\times\mathbb{P}^{3}\times\mathbb{P}^{1}$ in $\mathbb{P}^{23}$; thus we get a projective variety isomorphic to $X$ and embedded in a single projective space $\mathbb{P}^{19}=<X>\subset\mathbb{P}^{23}$.

i7 : s = segre X;

o7 : MultihomogeneousRationalMap (rational map from 4-dimensional subvariety of PP^2 x PP^3 x PP^1 to PP^19)
i8 : X' = projectiveVariety image s

o8 = X'

o8 : ProjectiveVariety, 4-dimensional subvariety of PP^19
i9 : (dim X', codim X', degree X')

o9 = (4, 15, 34)

o9 : Sequence
i10 : ? X'

o10 = 4-dimensional subvariety of PP^19 cut out by 102 hypersurfaces of
      degree 2

See also

Ways to use projectiveVariety :

For the programmer

The object projectiveVariety is a method function with options.