chowClass -- Finds the (fundamental) class of a subscheme in the Chow ring of the ambient space

Synopsis

• Usage:

  chowClass(IX)

  chowClass(IX,A)

• Inputs:
  • IX, an ideal, an ideal in the multi-graded coordinate ring 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:
  • Strategy (missing documentation) => ..., default value "multidegree", multidegree, using "prob" uses a probabilistic method which is sometimes faster on large examples
• Outputs:
  • isMultHom, a ring element, the class [X] in A where X is the subscheme associated to IX

Description

i1 : R=makeProductRing({6})

o1 = R

o1 : PolynomialRing

i2 : x=gens(R)

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

o2 : List

i3 : J=ideal(x_0*x_2-x_4*x_5)

o3 = ideal(a*c - e*f)

o3 : Ideal of R

i4 : clX=chowClass(J,Strategy=>"prob")

o4 = 2H
       1

     ZZ[H ]
         1
o4 : ------
        7
       H
        1

i5 : clX2=chowClass(J,ring(clX))

o5 = 2H
       1

     ZZ[H ]
         1
o5 : ------
        7
       H
        1

i6 : clX==clX2

o6 = true