Hi! My background is in (complex) algebraic geometry. For the first time I'm considering varieties defined over $\mathbb{R}$ and I have some very basic questions about these. In particular I'm trying to understand MMP for real surfaces.

Let $X$ be a smooth projective surface over $\mathbb{R}$. We can consider its complexification $X_{\mathbb{C}}$, smooth complex projective surface.

Then Cartier divisors on $X$ correspond to Cartier divisors on $X_{\mathbb{C}}$ that are fixed by conjugation. Is intersection theory on $X$ just inherited from intersection theory on $X_{\mathbb{C}}$ via this identification? Is this the right way to look at intersection theory on $X$?

What is the relation between the canonical divisors $K_X$ and $K_{X_{\mathbb{C}}}$? Is it true that $K_{X_{\mathbb{C}}}=\overline{K_{X_{\mathbb{C}}}}$, the divisor defined by conjugated equations and it thus corresponds to the divisor $K_X$ on $X$?

Also, how to define blow-ups of points on $X$? The definition in Hartshorne cap II.7 works but I imagine one can find an easier definition.

Whatever reference about these and related basic facts would be appraciated.