# Map of Grassmannians associated with a Veronese embedding

I'm quite sure this should be classically known, however I am not an expert on the topic and I was unable to find a precise reference in the huge literature concerning Veronese embeddings and Grassmannians. Any pointer to relevant books or papers will be highly appreciated.

Let $$V$$ be a finite-dimensional vector space (over $$\mathbb{C}$$, say) and let $$v_n \colon \mathbb{P}(V) \longrightarrow \mathbb{P}(S^n V)$$ be the usual $$n$$th Veronese embedding.

If $$\ell$$ is a line in $$\mathbb{P}(V)$$, then $$v_n(\ell)$$ is a rational normal curve of degree $$n$$, that will be contained in precisely one $$n$$-plane $$\Pi_{\ell} \subset \mathbb{P}(S^n V)$$. Then we can define a morphism of projective Grassmannians $$\psi_n \colon \mathbb{G}(1, \, \mathbb{P}(V)) \longrightarrow \mathbb{G}(n, \, \mathbb{P}(S^nV)),$$ given by $$\psi_n(\ell) :=\Pi_{\ell}$$.

Question. Is $$\psi_n$$ an embedding?

• Let $\mathbb{C}^2$ be the vector space associated to $l$. Isn'it that $\mathbb{P}(S^n \mathbb{C}^2) \cap v_n(\mathbb{P}(V)) = v_n(l)$? – Libli Dec 13 '18 at 10:23

I don't know a reference, but here is a simple argument. Note that $$G(2,V)$$ (let me use linear notation) is a homogeneous space for $$GL(V)$$: $$G(2,V) = GL(V)/P_2,$$ where $$P_2$$ is a parabolic. If $$e_1,\dots,e_N$$ is the basis of $$V$$, we can take $$P_2$$ to be the stabilizer of the point $$p_1 := [e_1 \wedge e_2] \in \mathbb{P}(\wedge^2V).$$ Note that $$e_1 \wedge e_2$$ is the highest weight vector with weight $$\epsilon_1 + \epsilon_2 = \omega_2$$ (the second fundamental weight of $$GL(V)$$).
The map $$\psi_n$$ is $$GL(V)$$-equivariant, and takes $$[e_1 \wedge e_2]$$ to $$p_n := [(e_1^n) \wedge (e_1^{n-1}e_2) \wedge \dots \wedge (e_1e_2^{n-1}) \wedge (e_2^n)].$$ It is easy to check that this is a highest vector with weight $$n\epsilon_1 + ((n-1)\epsilon_1 + \epsilon_2) + \dots (\epsilon_1 + (n-1)\epsilon_2) + n\epsilon_2 = \binom{n+1}{2}\omega_2$$ (it corresponds to an irreducible subrepresentation $$V_{\binom{n+1}{2}\omega_2} \subset \wedge^{n+1}(S^nV)$$), and its stabilizer is the same parabolic subgroup $$P_2$$. It follows that $$\psi_n$$ is an isomorphism onto the orbit of the point $$p_n$$, in particular it is an embedding.
• And both are special cases of the Borel-Weil-Tits embedding of $G/P_2$ into projective space of its geometric quantization, when endowed with its symplectic structure as coadjoint orbit of $\omega_2$, resp. ${n+1\choose 2}\omega_2$. – Francois Ziegler Dec 12 '18 at 21:31