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

- Usage:
    `projectiveVariety I`
- Inputs:
  - I, an ideal, a homogeneous ideal in a polynomial ring $R$ with the $\mathbb{Z}^n$-grading where the degree of each variable is a standard basis vector, that is, $R$ is the homogeneous coordinate ring of a product of $n$ projective spaces $\mathbb{P}^{k_1}\times\mathbb{P}^{k_2}\times\cdots\times\mathbb{P}^{k_n}$
- Optional inputs:
  - MinimalGenerators => ..., default value true, whether to trim the ideal (intended for internal use only)
  - Saturate => ..., default value true, whether to compute the multi-saturation of the ideal (intended for internal use only)
- Outputs:
  - a multi-projective variety, the projective subvariety of $\mathbb{P}^{k_1}\times\mathbb{P}^{k_2}\times\cdots\times\mathbb{P}^{k_n}$ defined by $I$

## 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 = segreEmbedding X;

    o7 : MultirationalMap (rational map from X to PP^19)

    i8 : X' = 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

- segreEmbedding(MultiprojectiveVariety) -- the Segre embedding of the variety
- dim(MultiprojectiveVariety) -- the dimension of the variety
- codim(MultiprojectiveVariety) -- the codimension of the variety
- degree(MultiprojectiveVariety) -- the degree of the variety
- singularLocus(MultiprojectiveVariety) -- the singular locus of the variety
- point(MultiprojectiveVariety) -- pick a random rational point on a multi-projective variety

## Ways to use projectiveVariety:

- `projectiveVariety(Ideal)`
- `projectiveVariety(Matrix)`
- `projectiveVariety(Ring)`
- `projectiveVariety(RingElement)`
- projectiveVariety(List,List,Ring) -- the Segre-Veronese variety
- `projectiveVariety(ZZ,ZZ,Ring)` -- see projectiveVariety(List,List,Ring) -- the Segre-Veronese variety
- projectiveVariety(List,Ring) -- product of projective spaces
- `projectiveVariety(ZZ,Ring)` -- see projectiveVariety(List,Ring) -- product of projective spaces
- projectiveVariety(MultidimensionalMatrix) -- the multi-projective variety defined by a multi-dimensional matrix

## For the programmer

The object projectiveVariety is a method function with options.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/MultiprojectiveVarieties.m2:2510:0.