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

Abstract

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 ) > 0 \operatorname {dim}H^{0}(L)>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 ) > 0 \operatorname {dim}H^{0}(L)>0 .