Connectivity of P2C curves

Question

I have a question related to the document “plane algebraic curves” from Andreas Gathmann: At page 37 Remark 5.12 (topology of complex curves), it is written: let F be a smooth projective curve over C (complex numbers), and V(F) its set of point. V(F) is then a compact one-dimensional complex manifold. In contrast to the real case, V(F) is always connected, because if it had two connected components X1 and X2, X1 and X2 would be V(F1) and V(F2) and applying Bézout Theorem to F1 and F2, they would intersect in degF1 . degF2 points, which contradicts the fact that F has two components X1 and X2. What I miss is “ X1 and X2 would be V(F1) and V(F2) where F1 and F2 are two projective curves, thus Bézout being applicable.” Any help would be welcome!