Quaternionic veronese Embedding

Solution 1:

Veronese embedding is the restriction of Segre embedding on the diagonal, so let's talk about quaternionic Segre embeddings instead. Now I have to admit, it's not a real answer, just 2 remarks:

1. Segre embedding $\mathbb CP^{\infty}\times\mathbb CP^{\infty}\to\mathbb CP^{\infty}$ gives a structure of an H-space on $\mathbb CP^{\infty}$. But I don't think $\mathbb HP^{\infty}$ admits an H-space structure. (See also Hatcher. 4L.4.)
2. Ordinary Segre embedding $\mathbb P(V)\times\mathbb P(W)\to\mathbb P(V\otimes W)$ maps a pair of 1-dimensional subspaces to their tensor product. Now, tensor product of two (say, left) quaternionic vector spaces is not a quaternionic vector space. But one can take tensor product over complex numbers — it induces a map $\mathbb P(V)\times\mathbb P(W)\to Gr_2(V\otimes_{\mathbb C}W)$.