Description
This can be considered a shortcut for dualize tangentialChowForm(I,dim I -1).
Note that in characteristic 0 (or sufficiently large characteristic), the reflexivity theorem implies that if I' == dualVariety I then dualVariety I' == I. Below, we verify the reflexivity theorem for the Veronese surface.
i1 : V = kernel veronese(2,2)
2 2 2
o1 = ideal (x - x x , x x - x x , x x - x x , x - x x , x x - x x , x -
4 3 5 2 4 1 5 2 3 1 4 2 0 5 1 2 0 4 1
------------------------------------------------------------------------
x x )
0 3
o1 : Ideal of QQ[x ..x ]
0 5
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i2 : time V' = dualVariety V
-- used 0.100439 seconds
2 2 2
o2 = ideal(x x - x x x + x x + x x - 4x x x )
2 3 1 2 4 0 4 1 5 0 3 5
o2 : Ideal of QQ[x ..x ]
0 5
|
i3 : time V == dualVariety V'
-- used 0.109759 seconds
o3 = true
|
In the next example, we verify that the discriminant of a generic ternary cubic form coincides with the dual variety of the 3-th Veronese embedding of the plane, which is a hypersurface of degree 12 in $\mathbb{P}^9$
i4 : F = first genericPolynomials({3,-1,-1},ZZ/3331)
3 2 2 3 2 2 2
o4 = a x + a x x + a x x + a x + a x x + a x x x + a x x + a x x +
0 0 1 0 1 3 0 1 6 1 2 0 2 4 0 1 2 7 1 2 5 0 2
------------------------------------------------------------------------
2 3
a x x + a x
8 1 2 9 2
ZZ
o4 : ----[a ..a ][x ..x ]
3331 0 9 0 2
|
i5 : time discF = ideal discriminant F;
-- used 0.0802816 seconds
ZZ
o5 : Ideal of ----[a ..a ]
3331 0 9
|
i6 : time Z = dualVariety(veronese(2,3,ZZ/3331),AssumeOrdinary=>true);
-- used 0.750444 seconds
ZZ
o6 : Ideal of ----[x ..x ]
3331 0 9
|
i7 : discF == sub(Z,vars ring discF) and Z == sub(discF,vars ring Z)
o7 = true
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