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;
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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}}];
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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
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i4 : X = projectiveVariety I
o4 = X
o4 : ProjectiveVariety, 4-dimensional subvariety of PP^2 x PP^3 x PP^1
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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
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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
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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)
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i8 : X' = image s
o8 = X'
o8 : ProjectiveVariety, 4-dimensional subvariety of PP^19
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i9 : (dim X', codim X', degree X')
o9 = (4, 15, 34)
o9 : Sequence
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i10 : ? X'
o10 = 4-dimensional subvariety of PP^19 cut out by 102 hypersurfaces of
degree 2
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