I am reading Artin’s notes “Lipman’s Proof of Resolution of Singularities for Surfaces” from the book “Arithmetic Geometry”. I am very confused by the proof of Lemma $ 6.5.$ (I am formulating it below in a little bit different way than it appears in the text)

Lemma 6.5: Let $ (A,\mathfrak m, k)$ be a normal complete excellent ring of dimension $ 2$ that defines a rational double point (rational Gorenstein singularity). Denote by $ X$ the blow-up of the unique closed point in Spec $ A$ . Assume that the exceptional divisor $ E$ is equal to $ 2C$ , where $ C$ is a line in $ \mathbb P^2_k$ . And let $ X’ \to X$ be a sequence of blow-ups in closed points $ p_1, \dots, p_n$ , s.t. $ X’$ is regular at evert point of the strict transform $ C’$ , then $ \Sigma_{i} [k(p_i):k]=3$ .

The key step is to compute $ \deg_C \mathcal O_X(-C)|_C$ . Artin claims that it is equal to $ -1$ , but I don’t understand his argument.

In our case, since $ 2C$ is isomorphic to a double line in $ \mathbb P^2$ , the degree is the same as for such a line, i.e., $ [-C,C]=-1$ .

How could one put this into a rigorous argument? It is not clear how to relate $ \deg_C \mathcal O_C(-C)$ with this immersion since $ \mathcal O_C(-C)^{\otimes 2} \neq \mathcal O_C(-E)$ .

P.S. By a rational singularity I mean that for any normal modification $ f:X \to Spec A$ we have $ H^1(X,\mathcal O_X)=0$ . If $ A$ is also Gorenstein, it is called rational double point. The latter condition is equivalent to $ \dim_k \mathfrak m/\mathfrak m^2 \leq 3$ .

P.S.2. In the formulation of Lemma $ E$ should be equal to a double line $ 2C$ with respect to the natural immersion $ X \to \mathbb P^2_A$ defined by the sheaf $ \mathcal O_X(-E)$ .