We classify nonconstant morphisms $\mathbb{ P}^m \to G/P$ for $m \le 4$ when $G = SL(n,\mathbb{ C})$ (type~$A$) for a minimal parabolic subgroup $P$. Using the Borel presentation of cohomology and explicit Schubert intersection identities, we show that there is no nonconstant morphism $\mathbb{ P}^2 \to G/B$; for minimal parabolic subgroup $P_{ \alpha_i}$, there are no nonconstant morphisms $\mathbb{ P}^3 \to G/P_{ \alpha_i}$ when $i \in \{ 1, n-1\}$, while such morphisms exist for $1 < i < n-1$; and, after correcting an earlier error (pointed out by Yanjie Li), we give an elementary proof that there is no nonconstant morphism $\mathbb{ P}^4 \to G/P_{ \alpha_i}$ for any minimal parabolic subgroup. The proofs are elementary and cohomological.
Categories: math.AG math.CO math.RT