TITLE: Image of the Veronese Embedding
QUESTION [7 upvotes]: Is the image of the general Veronese embedding ever contained in a hyperplane of $P^{n}$? I'm guessing no, but I can't prove it.

REPLY [15 votes]: 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.