Image of the Veronese Embedding
No. To prove it, imagine what it would mean for the image to be contained in a hyperplane: this would mean that some non-zero linear combination of the degree $d$ monomials vanished identically, which is to say, that there is some non-zero degree $d$ homogeneous equation which vanishes identically on $\mathbb P^n$. Hopefully you can convince yourself that this is not possible.