Virtual Euler Characteristic of a Curve in Brunella's paper

Question

I'm referring to

http://verbit.ru/MATH/Brunella/smf_bull_127_289-305.pdf

at pages 293 and 294. We are given two equations for a curve $C$ and a foliation $\mathcal{F}$ over a smooth algebraic surface $X$. The first one is when $C$ is invariant by $\mathcal{F}$: $$ c_1(T\mathcal{F})\cdot C = \mathcal{X}(C)-Z(\mathcal{F},C) $$ where $\mathcal{X}(C)$ is the virtual Euler characteristic of $C$ and $Z(\mathcal{F},C)$ is the sum of multiplicities of the singularities of $\mathcal{F}$ along $C$. The other equation is $$ c_1(T\mathcal{F})\cdot C=C^2-tang(\mathcal{F},C) $$

Now, proceeding to Lemma 1, we conclude that, if $C$ is $\mathcal{F}$-invariant, and supposing $Z(\mathcal{F},C)=0$, then we get $$ c_1(T\mathcal{F})\cdot C = 2 - Z(\mathcal{F},C). $$

Now, it's not pointed out anywhere, but this is possible if $\mathcal{X}(C)=2$. I could not see why this is true by myself, and would be pleased if someone could explain that to me.

Thanks in advance.

Answer 1

$C$ is rational, then $\chi(C)=2$.

For any projective curve in a surface, $\chi(C)=2 - 2p_a(C)$, $p_a(C)$ is the arithmetic genus of $C$. We have that $p_a(C) =0$ if, and only if, $C$ is a tree of smooth rational curves.

See BPV's Compact Complex Surfaces.