cycleClass -- determine the expression of the class of a cycle as a linear combination of Schubert classes

Synopsis

• Usage:

  cycleClass C

• Inputs:
  • C, an embedded projective variety, a subvariety of GG$(k, n)$ representing a cycle of pure codimension $m$ in the Grassmannian of $k$-dimensional subspaces of $\mathbb{P}^n$
• Outputs:
  • a ring element, the expression of the class of the cycle as a linear combination of Schubert classes

Description

For the general theory on Chow rings of Grassmannians, see e.g. the book 3264 & All That - Intersection Theory in Algebraic Geometry, by D. Eisenbud and J. Harris.

i1 : G = GG(ZZ/33331,2,5);

o1 : ProjectiveVariety, GG(2,5)

i2 : C = schubertCycle({3,2,1},G);

o2 : ProjectiveVariety, threefold in PP^19 (subvariety of codimension 6 in G)

i3 : cycleClass C

o3 = s
      3,2,1

o3 : ZZ[s     , s     , s     ]
         3,3,0   3,2,1   2,2,2

i4 : C' = C + schubertCycle({2,2,2},G);

o4 : ProjectiveVariety, threefold in PP^19 (subvariety of codimension 6 in G)

i5 : cycleClass C'

o5 = s      + s
      3,2,1    2,2,2

o5 : ZZ[s     , s     , s     ]
         3,3,0   3,2,1   2,2,2

See also

• schubertCycle -- take a random Schubert cycle