Question 4.7
Given
a^{1},
a^{2}, ...,
a^{s}
in
C_{n,k} with

a^{1}+
a^{2}+...+
a^{s}
equal to the dimension of
M^{q}_{k,n}
(which is
qn+
k(
n
k)), how many
maps
M in
M^{q}_{k,n} satisfy
M(t_{i}) lies in
Y_{ai} F.^{i}
for each i = 1, 2, ..., s ,
where
t_{1},
t_{2}, ...,
t_{s}
are general points in
P^{1} and
F.^{1},
F.^{2}, ...,
F.^{s} are general flags ?
Theorem 4.8 ([
So6, Theorem 1.1])
Let
q be a nonnegative integer and
n>
k>0 and set
N:=dim
M^{q}_{k,n}.
Then there exist
t_{1},
t_{2}, ...,
t_{N} in
RP^{1} and
(
n
k)planes
L_{1},
L_{2}, ...,
L_{N} in
R^{n} such that there are
exactly
d(
q;
k,
n) maps
M in
M^{q}_{k,n}(
C) satisfying
M(t_{i}) meets L_{i} nontrivially
for each i = 1, 2, ..., N ,
and all of them are real.
As with the classical Schubert calculus, the question of whether the general
quantum Schubert calculus is fully real remains open.