Let $X$ be a smooth projective connected surface over $\mathbb{C}$. It is a consequence of Bertini's theorem that $X$ contains, for every $g\in \mathbb{N}$, a smooth projective connected curve of genus at least $g$.
I am looking for a direct elementary proof of this fact, i.e., a proof I can explain to someone who just started doing algebraic geometry. (I am looking to avoid sections of line bundles, for example.)
I tried choosing an embedding into a projective space and to use intersections of $X$ with hypersurfaces, but how can I guarantee the smoothness and the increase of genus using, say, Riemann-Hurwitz?