I am currently following an introductory course in algebraic geometry (I am 5 chapters in 'An invitation to algebraic geometry') and stumbled on the following problem (we denote $\mathbb{P}^n$ for the standard $n$-dimensional projective space over $\mathbb{C}$):
Let $\nu_d : \mathbb{P}^n \mapsto \mathbb{P}^N$ be the $d$-Veronese map, where $N = \begin{pmatrix} n+d \\ d \end{pmatrix} - 1$ and $d > 0$. Let $g \in \mathbb{C}[x_0 , ... , x_n]$ be a homogeneous polynomial of degree $d$. Then prove that there exists a hyperplane $H \subset \mathbb{P}^N$ such that $\nu_d (\mathbb{P}^n) \cap H = \nu_d(\mathbb{V}(g))$.
An easy observation I was able to make was that it works for $d=1$, since in that case the Veronese map is the identical map and $g$ has degree 1, so we can take $H = \mathbb{V}(g)$.
Otherwise I don't know how to start solving this problem. If someone could help me get started, that would be very helpful.
Thank you in advance.