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

Synopsis

• Usage:

  segre(IX,IY)

  segre(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 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({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 : segre(X,D)

          3 3     3 2     2 3
o5 = - 10H H  + 3H H  + 3H H
          1 2     1 2     1 2

     ZZ[H ..H ]
         1   2
o5 : ----------
        4   4
      (H , H )
        1   2

i6 : A = makeChowRing(R)

o6 = A

o6 : QuotientRing

i7 : s = segre(X,D,A)

          3 3     3 2     2 3
o7 = - 10H H  + 3H H  + 3H H
          1 2     1 2     1 2

o7 : A