Here is the situation: $S$ is a nonsingular complex projective surface, and $C=X\cup_AY\subset S$ is a uninodal curve of compact type: it is obtained by glueing two nonsingular curves $X,Y\subset S$ at a single node $A$.
[If this helps (or is needed), I obtained $C$ as the special fiber of a family of smooth curves $\pi:S\to B$, where $B=\textrm{Spec }\mathbb C[[t]]$ and the generic fiber $S_\eta$ is a nonsingular curve.]
Question. Why is $X.C=0$?
If $X$ was a fiber of $\pi$ as well, we would be done. But it isn't. Moreover, the assertion $X.C=0$ looks strange to me, because (please, correct me here because I feel like I am dramatically wrong)
it means that $X$ is linearly equivalent to $X+Y$. But $X-(X+Y)=-Y$ is not equivalent to $0$!
Moreover, how should I interpret the fact that $X.X=-A$? (This follows by the claim because $0=X.C=X.(X+Y)=X.X+X.Y=X.X+A$).
Thank you.