Lower bound of degree of ruled surface $S\subset\mathbb{P}^n$.

Question

I have a question of Complex Algebraic surface in Beauville.
Let $S\subset\mathbb{P}^n$ be a surface of degree $d$ lying in no hyperplane.

1. Show that $d\geq 2 n-2$ if $S$ is not ruled, and that $K_S\equiv 0$ if equality holds.
2. Show that $d\geq n+q-1$ if $S$ is ruled.

I finish the first problem. For the second problem, I can only prove the partial case as follows.

Let $H$ be the hyperplane section of $S$ in $\mathbb{P}^n$ and $g$ be the genus of $H$. Consider the global section of exact $$0\to\mathscr{I}_S(1)\to\mathscr{O}_{\mathbb{P}^n}(1)\to\mathscr{O}_S(1)\to 0.$$ Since the global section of $\mathscr{I}_S(1)$ consists of hyperplane sections vanishing on $S$ but $S$ lies in no hyperplane, $H^0(\mathbb{P}^n,\mathscr{I}_S(1))=0$. Hence $$\dim H^0(S,\mathscr{O}_S(H))\geq\dim H^0(\mathbb{P}^n,\mathscr{O}_{\mathbb{P}^n}(1))=n+1. \tag{1}$$ The exact sequence $$0\to\mathscr{O}_S\to\mathscr{O}_S(H)\to\mathscr{O}_H(H)\to 0,$$ induces the long exact seuqence \begin{align} 0\to & \mathbb{C}\to H^0(S,\mathscr{O}_S(H))\to H^0(H,\mathscr{O}_H(H))\to H^1(S,\mathscr{O}_S)\to H^1(S,\mathscr{O}_S(H))\to H^1(H,\mathscr{O}_H(H)) \\ \to & H^2(S,\mathscr{O}_S)\to H^2(S,\mathscr{O}_S(H))\to H^2(H,\mathscr{O}_H(H)). \\ \end{align} Hence $h^0(H,\mathscr{O}_H(H))\geq n$. By Riemann-Roch theorem on $\mathscr O_S(H)$, $$h^0(H)-h^1(H)+h^2(H)=\dfrac{1}{2}H.(H-K)+\chi(\mathscr{O}_S)=d-g+2-q.$$ Since $S$ is ruled, $p_1(S)=0$ by Enriques theorem. By Serre duality, $h^2(S,\mathscr{O}_S)=h^0(S,K_S)=0$. Also $H^2(H,\mathscr{O}_H(H))=0$, since $\dim H=1$. Hence $h^2(H)=0$. Combine with (1), $$d\geq n+q-1+g-h^1(H).$$

For the case of $K_S.H<0$. By Serre duality on curve $H$ and adjunction formula, $$h^1(H,\mathscr{O}_H(H))=h^0(H,K_H-\mathscr{O}_H(H))=h^0(H,\mathscr{O}_H(K_S))=0$$ since $\text{deg}_H K_S=H.K_S<0$. By the long exact sequence in above, I have $q\geq h^0(S,\mathscr{O}_S(H))$. But we have $g\geq q$ by Albanese map of $S$ sent $H$ to the curve with genus $q$. Hence $d\geq n+q-1$ in this case.

But I can't solve the case when $K_S.H\geq 0$. Hope someone can answer or provide me some ideas.