On sectional genus of quasi-polarized 3-folds

Abstract

Let X X be a smooth projective variety over C \mathbb {C} and L L a nef-big (resp. ample) divisor on X X . Then ( X , L ) (X,L) is called a quasi-polarized (resp. polarized) manifold. Then we conjecture that g ( L ) ≥ q ( X ) g(L)\geq q(X) , where g ( L ) g(L) is the sectional genus of L L and q ( X ) = dim ⁡ H 1 ( O X ) q(X)=\operatorname {dim}H^{1}(\mathcal {O}_{X}) is the irregularity of X X . In general it is unknown whether this conjecture is true or not, even in the case of dim ⁡ X = 2 \operatorname {dim}X=2 . For example, this conjecture is true if dim ⁡ X = 2 \operatorname {dim}X=2 and dim ⁡ H 0 ( L ) > 0 \operatorname {dim}H^{0}(L)>0 . But it is unknown if dim ⁡ X ≥ 3 \operatorname {dim}X\geq 3 and dim ⁡ H 0 ( L ) > 0 \operatorname {dim}H^{0}(L)>0 . In this paper, we prove g ( L ) ≥ q ( X ) g(L)\geq q(X) if dim ⁡ X = 3 \operatorname {dim}X=3 and dim ⁡ H 0 ( L ) ≥ 2 \operatorname {dim}H^{0}(L)\geq 2 . Furthermore we classify polarized manifolds ( X , L ) (X,L) with dim ⁡ X = 3 \operatorname {dim}X=3 , dim ⁡ H 0 ( L ) ≥ 3 \operatorname {dim}H^{0}(L)\geq 3 , and g ( L ) = q ( X ) g(L)=q(X) .