On open algebraic surfaces of general type whose log canonical maps composed of a pencil

2023/02/19 by Zhao Hang, Zhao, Hang
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics #Geometric and Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2302.09619

openalex publication_date 2023/02/19 · openalex created_date 2023/02/23 · openalex updated_date 2026/07/28

Abstract

Let (S,D) be a minimal log pair of general type with S a smooth projective surface and D a simple normal corssing reduced divisor on S. We assume that its log canonial linear system |K_S+D| is composed of a penciel, let f\colon S→ B be the fiberation induced by the linear system |K_S+D| and F be a general fiber of f. Let b (resp. g) be the genus of the base curve B (resp. general fiber F) and k=D⋅ F the intersection number. We show that 1. If k>0 and b≥ 2 then 2≤ g+k ≤ 3, when g+k=3 we have b=2 and h¹(S,K_S+D)=0. 2. Suppose p_a(D)≤ 2(l+q(S))+1-h^1,1(S) where l is the number of irreducible components of D, then we have g≤ 5 for p_g(S,D)≫ 0. Moreover if p_g(S)=0, then we have g≤ 3.

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