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SegreClasses :: projectiveDegree

projectiveDegree -- This method computes a single projective degree of a scheme X inside a scheme Y, where X,Y are subschemes of some product of projective spaces

Synopsis

Description

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
i2 : x = gens(R)

o2 = {a, b, c, d, e, f, g, h}

o2 : List
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
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
i7 : h=A_0^2*A_1^2

      2 2
o7 = H H
      1 2

o7 : A
i8 : pdh=projectiveDegree(X,D,h)

o8 = 3
i9 : (sum pd)_h==pdh

o9 = true

Ways to use projectiveDegree :