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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.

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up vote 3 down vote accepted

The degree of $C$ in $\mathbb P^N$ is the intersection number of a hyperplane with $C$, or equivalently, the degree (as divisor on $C$) of the restriction of a hyperplane to $C$. In terms of invertible sheaf, a hyperplane corresponds to $O_{\mathbb P^N}(1)$ and its restriction to $C$ is, by construction, $nK_C$. So the degree of $C$ in $\mathbb P^N$ is just the degree on $C$ of $nK_C$, which is $n(2g-2)=2n(g-1)$ by Riemann-Roch.

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