I am studying Beauville's book "Complex Algebraic Surfaces". At page 2 he defines the intersection form (.) on the Picard group of a surface. For $L, L^\prime \in Pic(S)$
$$(L.L^\prime)=\chi(\mathcal{O}_S)-\chi(L^{-1})-\chi(L^{\prime-1})+\chi(L^{-1}\otimes L^{\prime-1})$$
Why the self-intersection (i.e $(H.H)=H^2$) of an hyperplane section $H$ on $S$ is always positive?
This self-intersection is exactly the degree of $S$.
Concretely, choose $H$ and $H'$ in general position, then $S \cap H$ and $S \cap H'$ are two curves on $S$, and they intersect in some number of points. This already shows that the intersection is non-negative. The fact that it is positive is a general fact about projective varieties: projective varieities of complementary dimension always have a positive number of intersection points. (Apply this to $S$, which is of dimension $2$, and $H \cap H'$, which is a linear subspace of codimension $2$, i.e. of complementary dimension to $S$.)