segreDimX -- This method computes the dimension X part of the Segre class of a scheme X inside a scheme Y, where X,Y are subschemes of some product of projective spaces

Synopsis

• Usage:

  segreDimX(IX,IY,A)

• Inputs:
  • IX, an ideal, a multi-homogeneous ideal defining a closed subscheme of \PP^{n_1}x...x\PP^{n_m}; makeProductRing builds the graded coordinate ring of \PP^{n_1}x...x\PP^{n_m}.
  • IY, an ideal, a multi-homogeneous ideal defining a closed subscheme of \PP^{n_1}x...x\PP^{n_m}; makeProductRing builds the graded coordinate ring of \PP^{n_1}x...x\PP^{n_m}.
  • A, a quotient ring, the Chow ring of \PP^{n_1}x...x\PP^{n_m}. This ring can be built by applying makeChowRing to the coordinate ring of \PP^{n_1}x...x\PP^{n_m}.
• Optional inputs:
  • Verbose (missing documentation) => a Boolean value, default value false,
• Outputs:
  • s, a ring element, the dimension X part of the Segre class of the subscheme X defined by IX in the subscheme Y defined by IY as a class in the Chow ring of \PP^{n_1}x...x\PP^{n_m}.

Description

i1 : R = makeProductRing({2,2})

o1 = R

o1 : PolynomialRing

i2 : x = gens(R)

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

o2 : List

i3 : Y = ideal(random({2,2},R));

o3 : Ideal of R

i4 : X = Y+ideal(x_0*x_3+x_1*x_4);

o4 : Ideal of R

i5 : A = makeChowRing(R)

o5 = A

o5 : QuotientRing

i6 : time s = segreDimX(X,Y,A)
     -- used 0.24438 seconds

       2             2
o6 = 2H  + 4H H  + 2H
       1     1 2     2

o6 : A

i7 : time segre(X,Y,A)
     -- used 0.168976 seconds

        2 2     2         2     2             2
o7 = 12H H  - 6H H  - 6H H  + 2H  + 4H H  + 2H
        1 2     1 2     1 2     1     1 2     2

o7 : A