Why is , $ \mathcal{O} (1) $ restricted to a curve $ C $, of degree $ d $?

Question

Let $ C \subset \mathbb{P}^2 $ be a smooth curve defined by a homogeneous polynomial $ f $ of degree $ d $.

How to establish that the line bundle $ \mathcal{O} (1) $ restricted to $ C $ is of degree $ d $ ?

Thanks in advance for your help.

Answer 1

I don't understand if the answer to this question is clear or no, but it seems to me that it isn't.

First of all, note that in our situation talking about Cartier or Weil divisors is the same, so I suggest to look at the latter which is a bit more intuitive. A Weil divisor is basically a subscheme of codimension $1$ (plus all the hypotheses which are needed). The group of divisors is $Div(X)$. On the other hand we have line bundles, or better invertible sheaves. The group they form is $Pic(X)$. As you said there is a map $Div(X)\to Pic(X)$. Moreover if $Cl(X)$ is the quotient group of $Div(X)$ by linear equivalence, then $Cl(X)\cong Pic(X)$. (Everything here is in Hartshorne).

Now, the invertible sheaf $\mathcal{O}(1)\in Pic(\mathbb{P}^2)$ corresponds, under the above isomorphism, to the class of a hyperplane $[H]\in Cl(\mathbb{P}^2)$ (this is basically the definition of $\mathcal{O}(1)$). The restriction of the invertible sheaf $\mathcal{O}(1)$ to $C$ is an invertible sheaf on $C$, i.e. $\mathcal{O}(1)|_C\in Pic(C)$, and its (inverse) image in $Cl(C)$ is the ''restriction" $[H|_C]$, which is nothing more than the (class of the) intersection of $C$ and $H$. Here you may probably want to mess up with the definition of restriction map to convince yourself that it is true. Since $C$ is defined by a degree $d$ polynomial and $H$ is defined by a linear polynomial, their intersection consists of at most $d$ points, counted with the right multiplicities, and the sum of their multiplicities is $d$. This is basically Bezout theorem. Hence $H|_C=\sum_{p\in H\cap C}mult(p)\cdot p$ and its degree is $d$. Note that here we are working with a fix hyperplane $H$, but in projective space all hyperplanes are linear equivalent each other and we know that the degree is invariant on each linear equivalence class. Finally the degree of the invertible sheaf $\mathcal{O}(1)|_C$ is also $d$ by definition.