Let (S,H) be a polarized surface (S is a complex connected projective manifold of dimension two and H is an ample divisor on S). The following classes are introduced: \({\mathcal A}=\{{\mathbb{P}}^ 2,{\mathcal O}_{{\mathbb{P}}^ 2}(r)),\quad r=1,2\}, {\mathcal B}=\{(S,H)| S={\mathbb{P}}^ 1-bundle,\quad [H]_{| f}={\mathcal O}_{{\mathbb{P}}^ 1}(1),\quad for\quad any\quad fibre\quad f\}, {\mathcal C}=\{(S,H)\quad conic\quad bundle\}, {\mathcal D}\quad =\{(S,H)| H\equiv -K_ S,\quad an\quad anticanonical\quad divisor\}.\) The main result is the following theorem. For any polarized surface (S,H) either (i) \(K_ S+H\) is ample, (ii) (S,H)\(\in {\mathcal A}\cup {\mathcal B}\cup {\mathcal C}\cup {\mathcal D}\), or (iii) S is not a minimal model, \((K_ S+H)^ 2>0, (K_ S+H)C\geq 0\) for any curve \(C\subset S\), with equality iff C is a (-1)-curve such that \(CH=1\). In this case, by contracting such curves via a birational morphism \(\eta\) one gets a polarized surface \((S',H'=\eta_*H)\) where \(K_{S'}+H'\) is ample. The main facts used in the proof are some arguments from Mori’s theory of extremal rays [S. Mori, Ann. Math., II. Ser. 116, 133-176 (1982)] and the following proposition. If (S,H)\(\not\in {\mathcal B}\), then \((K_ S+H)^ 2\geq 0,\) and equality holds iff (S,H)\(\in {\mathcal C}\cup {\mathcal D}\). Some applications are given. Let g be the arithmetic genus of H, \(p_ g\) the geometric genus of S and q its irregularity. \((1) g\geq \max \{0,q- p_ g\}\) with equality iff (S,H)\(\in {\mathcal A}\cup {\mathcal B}\); \((2) (S,H)\in {\mathcal B}\) iff \(K^ 2\!_ S=8(1-g);\) (3) If \(g=0\), then S is rational and (S,H)\(\in {\mathcal A}\cup {\mathcal B}\); if \(g=1\), then either (S,H)\(\in {\mathcal D}\) or (S,H)\(\in {\mathcal B}\) and \(q=1\); \((4) h^ 0(K_ S+H)\geq 1\) iff (S,H)\(\not\in {\mathcal A}\cup {\mathcal B}\) and if equality holds then either (S,H)\(\in {\mathcal D}\) or \(g=2\) and S is not of general type. Properties (1)-(3) extend classical results on hyperplane sections.