The bicanonical map of surfaces with p_g=0 and K² ≥ 7

1999/10/14 by Margarida Mendes Lopes, M. Mendes Lopes, Rita Pardini +3
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #math.AG

paper · pdf · doi:10.48550/arxiv.math/9910074

LaTeX 2e, 15 pages, to appear. Minor typos, change of title (title of the previous version: "A note on surfaces of general type with $p_g=0$ and $K^2\ge 7$") and added one example of a surface, showing that the main result is sharp

openalex publication_date 1999/10/14 · arxiv created 2000/09/18 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A minimal surface of general type with p_g(S)=0 satisfies 1≤ K²≤ 9 and it is known that the image of the bicanonical map \fie is a surface for K_S²≥ 2, whilst for K²_S≥ 5, the bicanonical map is always a morphism. In this paper it is shown that \fie is birational if K_S²=9 and that the degree of \fie is at most 2 if K_S²=7 or K_S²=8. By presenting two examples of surfaces S with K_S²=7 and 8 and bicanonical map of degree 2, it is also shown that this result is sharp. The example with K_S²=8 is, to our knowledge, a new example of a surface of general type with p_g=0.

Related

• The classification of surfaces with p_g=0, K²=6 and non birational bicanonical map
• The degree of the bicanonical map of a surface with p_g=0
• A survey on the bicanonical map of surfaces with p_g=0 and K²≥ 2
• A new family of surfaces with p_g=0 and K²=3
• The bicanonical map of surfaces with p_g=0 and K²≥ 7, II