Let $X\subset \mathbb{P}^n$ be a smooth subvariety of degree $e$ and dimension $m$. Consider the $d$-uple embedding $\mathbb{P}^n\hookrightarrow \mathbb{P}^N$ where $N={n+d \choose n}-1$. We know the degree the image of $\mathbb{P}^n$ is $d^n$.
Question: what is the degree of $X$ under the $d$-uple embedding. Is it $ed^m$?
Thanks!