On the Birationality of the Bicanonical Map of a Surface Section of a Threefold

Abstract

Let (ℳ, ℒ) be a polarized threefold of log-general type. The birationality of the bicanonical map of a smooth surface S ∈ |ℒ| is studied. This problem was previously considered and partially solved by the first and fourth author, who gave a satisfactory classification unless h¹(𝒪_ℳ) = 0 and p_g(S) = 3, 4, 5. This article focuses on the remaining cases which are the hardest, settling the problem.

Keywords:

• Adjunction theory
• Bicanonical map
• Log-general type
• Smooth complex polarized threefold
• Surface section
• Surface of general type
• Very ample line bundle

2000 Mathematics Subject Classification:

• Primary 14N30, 14J30, 14E05, 14C20
• Secondary 14J29, 14J10

Notes

Communicated by C. Pedrini.