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Let      X X       be a smooth projective surface over      C \mathbb {C}       and      L L       an ample Cartier divisor on      X X      . If the Kodaira dimension      κ  ( X ) ≤  1 \kappa (X)\leq 1       or      dim ⁡  H 0 ( L ) &gt; 0 \operatorname {dim}H^{0}(L)&gt;0      , the author proved      g ( L ) ≥  q ( X ) g(L)\geq q(X)      , where      q ( X ) = dim ⁡  H 1 ( O X ) q(X)=\operatorname {dim}H^{1}(\mathcal {O}_{X})      . If      κ  ( X ) ≤  1 \kappa (X)\leq 1      , then the author studied      ( X , L ) (X,L)       with      g ( L ) = q ( X ) g(L)=q(X)      . In this paper, we study the polarized surface      ( X , L ) (X,L)       with      κ  ( X ) = 2 \kappa (X)=2      ,      g ( L ) = q ( X ) g(L)=q(X)      , and      dim ⁡  H 0 ( L ) &gt; 0 \operatorname {dim}H^{0}(L)&gt;0      .

On polarized surfaces (𝑋,𝐿) with ℎ⁰(𝐿)>0, 𝜅(𝑋)=2, and 𝑔(𝐿)=𝑞(𝑋)