Let $n\geq 3$ be an integer. If we embed a connected curve $C$ (e.g. a stable curve) of genus $g$ in $\mathbb P^N$ by an $n$-canonical embedding, i.e. using the very ample linear system $|nK_C|$, we have that $N=(2n-1)(g-1)-1$. This is clear. But I do not see how to deduce that the degree of $C$ is $2n(g-1)$. This is equivalent to the assertion \begin{equation} g+\deg C=N, \end{equation} which I am not able to justify. Does anyone have any hint? Is it possible to use some adjunction formula argument even if we are not in the plane case?
Thanks.