Let $S \subset \mathbb{P}^{n}_{\mathbb{C}}$ a projective integral smooth surface lying in no hyperplane (I adopt the point of view of scheme). I will denote by $d$ its degree. The following questions come from Beauville's book on complex algebraic surface (Exercises VI.22 page $85$).
(a) I want to show $d \ge 2n - 2$ if $S$ is not ruled (i.e. not birationnaly equivalent to $C \times_{\mathbb{C}} \mathbb{P}^{1}_{\mathbb{C}}$ for $C$= a curve) and that $K_{S}$ (the canonical divisor of $S$) is principal if equality holds.
(b)I want to show $d \ge n+q -1$ if $S$ is ruled.
What I try :
(a)By Bertinni's theorem (Hartshorne Theorem $8.18$ page $179$) their exists a smooth hyperplane section $H$ of $S$. Since $S$ is not ruled, then $H.K_{S} \ge 0$ by Corollary $VI.18$ page $83$ of Beauville book on complex algebraic surfaces. Let $D$ be the divisor associated to $\mathcal{O}(H)_{|H}$. We have $d = H^{2} = deg(\mathcal{O}(H)_{|H}) = deg(D)$ by the definition of the degree. By adjunction formula, we have $deg(D) + H.K_{S} = (H+K_{S}).H = 2g(H) - 2$ so $d \leq 2(g(H) - 1)$ (with equality if $K_{S}$ is principal $K_{S} \equiv 0$). By Clifford inequality, we have $h^{0}(D) \leq \frac{1}{2}d +1$ if $D$ is special (see Hartshorne page $296$) so that $|D| \leq \frac{d}{2}$. I don't know what to do next.
$\\\\$(b) By Riemann Roch, $h^{0}(H) - h^{1}(H) = \frac{1}{2}H.(H - K_{S}) + \chi(\mathcal{O}_{S}) = $ (by adjunction formula) $ H^{2} -g(H) + 1 + 1 - q - h^{2}(\mathcal{O}_{S}) = $ because $S$ is ruled $ d - g(H) + 2 - q$ so that $d = g(H) - 2 + q + h^{0}(H) - h^{1}(H) \ge g(H) - 2+ q + n+1 - h^{1}(H) = q + n - 1 + g(H) - h^{1}(H) \ge q + n - 1$ because $h^{0}(H) \ge n+1$ and because $g(H) - h^{1}(H) \ge 0$. However, I don't know how to show $g(H) - h^{1}(H) \ge 0$
