For subschemes X, Y of ℙn1x...xℙnm this command computes a projective degree associated to h of a subscheme X in the subscheme Y as classes in the Chow ring of ℙn1x...xℙnm. The value returned is an integer. This method is faster if only one projective degree is needed.
i1 : R = makeProductRing({3,3})
o1 = R
o1 : PolynomialRing
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i2 : x = gens(R)
o2 = {a, b, c, d, e, f, g, h}
o2 : List
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i3 : D = minors(2,matrix{{x_0..x_3},{x_4..x_7}})
o3 = ideal (- b*e + a*f, - c*e + a*g, - c*f + b*g, - d*e + a*h, - d*f + b*h,
------------------------------------------------------------------------
- d*g + c*h)
o3 : Ideal of R
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i4 : X = ideal(x_0*x_1,x_1*x_2,x_0*x_2) o4 = ideal (a*b, b*c, a*c) o4 : Ideal of R |
i5 : A = makeChowRing(R) o5 = A o5 : QuotientRing |
i6 : pd = projectiveDegrees(X,D,A)
3 3 2 3 3 2 3 2 2 3
o6 = {10H H , 6H H , 6H H , 3H H , 3H H , 3H H }
1 2 1 2 1 2 1 2 1 2 1 2
o6 : List
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i7 : h=A_0^2*A_1^2
2 2
o7 = H H
1 2
o7 : A
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i8 : pdh=projectiveDegree(X,D,h) o8 = 3 |
i9 : (sum pd)_h==pdh o9 = true |