intersectionProduct -- A class in the Chow ring of the ambient space representing the Fulton-MacPherson intersection product of two schemes inside a variety

Synopsis

• Usage:

  intersectionProduct(IX,IV,IY)

  intersectionProduct(IX,IV,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}.
  • IV, 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 smooth complete intersection in \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:
  • intProd, a ring element, a class in the Chow ring A of \PP^{n_1}x...x\PP^{n_m} representing the Fulton-MacPherson intersection product of X with V in Y (where X is the scheme associated to IX, etc.).

Description

i1 : R = makeProductRing({3})

o1 = R

o1 : PolynomialRing

i2 : (x,y,z,w) = toSequence gens R

o2 = (a, b, c, d)

o2 : Sequence

i3 : Q = ideal(x*y-z*w)

o3 = ideal(a*b - c*d)

o3 : Ideal of R

i4 : L1 = ideal(x,w)

o4 = ideal (a, d)

o4 : Ideal of R

i5 : L2 = ideal(y,w)

o5 = ideal (b, d)

o5 : Ideal of R

i6 : intersectionProduct(L1,L2,Q,Verbose=>true)
      2 2
[Y]= H H , alpha= 2H  + 2H
      1 2           1     2
                       3 3
Projective degrees= {7H H }
                       1 2
         3 3
s(X,Y)= H H
         1 2
                              3
Segre pullback to diagonal = H
                              1
                2
Chern class = 2H  + 2H  + 1
                1     1

      3
o6 = H
      1

     ZZ[H ]
         1
o6 : ------
        4
       H
        1

i7 : intersectionProduct(L1,L1,Q)

o7 = 0