# Degrees of subvarieties of projective space

I've always thought of the degree of a subvariety of projective space as the degree of the divisor that defines the (given) embedding into projective space. It's been pointed out to me that this works only for curves. Now I'm confused: is there a similar characterization of the degree of a general subvariety of some projective space?

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Maybe the reason for your confusion is that on a curve the degree of a divisor is intrinsically defined, whereas in general the notion of degree of a divisor depends on the projective embedding of the variety. Nevertheless, as Allen says below, the relationship still holds in higher dimensions. –  Johan Mar 27 '10 at 18:28

If $X\subset \mathbb P^n$ is a subvariety of dimension $m$ embedded by a linear system $V \subset H^0(X,\mathcal O_X(D))$ then the degree of $X$ is equal to $D^m$.
I disagree with "this works only for curves". Say we've already defined degree for schemes up to dimension $n$. Then use your rule to define it for schemes of dimension $n+1$. Dr. "This works only for curves" is unwilling to go beyond the $n=0$ case, for some reason.