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Schubert2 :: map(FlagBundle,AbstractVariety,AbstractSheaf)

map(FlagBundle,AbstractVariety,AbstractSheaf) -- maps to projective bundles

Synopsis

Description

Accepts both Grothendieck-style and Fulton-style ℙ(E), but in the case a decision cannot be made based on ranks (i.e. when E has rank 2), defaults to Fulton-style notation (so ℙ(E) is the space of sub-line-bundles of E).

Does not check whether L is basepoint-free. Weird results are probably possible if L is not.
i1 : X = flagBundle({2,2}) --the Grassmannian GG(1,3)

o1 = X

o1 : a flag bundle with ranks {2, 2}
i2 : (S,Q) = bundles X

o2 = (S, Q)

o2 : Sequence
i3 : L = exteriorPower(2,dual S)

o3 = L

o3 : an abstract sheaf of rank 1 on X
i4 : P = flagBundle({5,1}) --Grothendieck-style PP^5

o4 = P

o4 : a flag bundle with ranks {5, 1}
i5 : f = map(P,X,L) -- Plucker embedding of GG(1,3) in PP^5

o5 = f

o5 : a map to P from X
i6 : H = last bundles P

o6 = H

o6 : an abstract sheaf of rank 1 on P
i7 : f^* (chern(1,H)) -- hyperplane section, should be sigma_1

o7 = H
      2,1

                         QQ[][H   , H   , H   , H   ]
                               1,1   1,2   2,1   2,2
o7 : --------------------------------------------------------------------
     (H    + H   , H    + H   H    + H   , H   H    + H   H   , H   H   )
       1,1    2,1   1,2    1,1 2,1    2,2   1,2 2,1    1,1 2,2   1,2 2,2
i8 : f_* chern(0,S) --expect 2 times hyperplane class since GG(1,3) has degree 2

o8 = 2H
       2,1

                               QQ[][H   , H   , H   , H   , H   , H   ]
                                     1,1   1,2   1,3   1,4   1,5   2,1
o8 : -------------------------------------------------------------------------------------------
     (H    + H   , H    + H   H   , H    + H   H   , H    + H   H   , H    + H   H   , H   H   )
       1,1    2,1   1,2    1,1 2,1   1,3    1,2 2,1   1,4    1,3 2,1   1,5    1,4 2,1   1,5 2,1