Suppose that $E\subseteq S$ is a $(-1)$-curve inside a non-singular complex projective surface. By a $(-1)$-curve $E$, I mean that $E\cong\mathbb P^1_\mathbb C$ and $E^2=-1$.
Now consider a field automorphism $\sigma\in\text{Aut}(\mathbb C)$ and the fibered product along the morphism of schemes $\text{Spec }(\sigma):\text{Spec }\mathbb C\longrightarrow\text{Spec }\mathbb C$: $$E^\sigma:=E\times_{\text{Spec }\mathbb C}\text{Spec }\mathbb C$$
Is it true that ${(E^\sigma)}^2=-1$? In other words, is the auto-intersection preserved by the action of $\text{Aut}(\mathbb C)$ on $E$?
Many thanks in advance.
The intersection product will be preserved for products $C\cdot D$, in the case where the intersection consists of reduced points. But these determine the intersection product in general (by the moving lemma).
So yes, it is true.