[Paper Review] The bicanonical map of surfaces with $p_g=0$ and $K^2\ge 7$, II

Margarida Mendes Lopes, Rita Pardini|ArXiv.org|Nov 13, 2001

Algebraic Geometry and Number Theory8 references4 citations

TL;DR

This paper classifies minimal complex surfaces of general type with $p_g=0$ and $K^2=7$ or $8$ whose bicanonical map has degree 2. It proves that such surfaces admit a genus 3 hyperelliptic fibration over $\mathbb{P}^1$, with the bicanonical involution restricting to the hyperelliptic involution on fibers. For $K^2=8$, the fibration is isotrivial with 6 double fibers; for $K^2=7$, it has 5 double fibers and one reducible fiber with two components.

ABSTRACT

We study the minimal complex surfaces of general type with $p_g=0$ and $K^2=7$ or 8 whose bicanonical map is not birational. In the paper 'The bicanonical map of surfaces with $p_g=0$ and $K^2\ge 7$' we have shown that if $S$ is such a surface, then the bicanonical map has degree 2. Here we describe precisely such surfaces showing that there is a fibration $f\colon S o \pp^1$ such that: i) the general fibre $F$ of $f$ is a genus 3 hyperelliptic curve; ii) the involution induced by the bicanonical map of $S$ restricts to the hyperelliptic involution of $F$. Furthermore, if $K^2_S=8$, then $f$ is an isotrivial fibration with 6 double fibres, and if $K^2_S=7$, then $f$ has 5 double fibres and it has precisely one fibre with reducible support, consisting of two components.

Motivation & Objective

• To classify minimal complex surfaces of general type with $p_g=0$ and $K^2=7$ or $8$ for which the bicanonical map is not birational.
• To determine the geometric structure of such surfaces when the bicanonical map has degree 2.
• To characterize the fibration structure and the behavior of the bicanonical involution on fibers.
• To establish the existence and properties of double fibers and reducible fibers in the fibration.

Proposed method

• Use of the bicanonical map and its degree to classify surfaces with $p_g=0$ and $K^2=7$ or $8$.
• Application of involution theory on surfaces to analyze the action of the bicanonical involution.
• Construction of a fibration $f: S \to \mathbb{P}^1$ with general fiber a genus 3 hyperelliptic curve.
• Use of the adjunction formula and double cover theory to relate canonical divisors on quotient surfaces.
• Analysis of the restriction of the bicanonical system to fibers using cohomological techniques and Serre duality.
• Application of Zariski's Lemma and the theory of $1$-connected divisors to study base points and fiber decompositions.

Experimental results

Research questions

• RQ1 What is the structure of the bicanonical map for minimal surfaces of general type with $p_g=0$ and $K^2=7$ or $8$ when it is not birational?
• RQ2 Does the bicanonical involution on such surfaces restrict to the hyperelliptic involution on the fibers of a fibration?
• RQ3 What are the properties of the fibration $f: S \to \mathbb{P}^1$ induced by the bicanonical system?
• RQ4 How do the number and type of double fibers and reducible fibers differ between the $K^2=7$ and $K^2=8$ cases?
• RQ5 Is the canonical class $K_S$ ample in these cases, and what does this imply for the geometry of the surface?

Key findings

• The bicanonical map has degree 2 for minimal surfaces with $p_g=0$, $K^2=7$ or $8$, and the bicanonical involution induces the hyperelliptic involution on the general fiber.
• For $K^2_S = 8$, the fibration $f: S \to \mathbb{P}^1$ is isotrivial and has exactly 6 double fibers.
• For $K^2_S = 7$, the fibration has 5 double fibers and exactly one fiber with reducible support, consisting of two components.
• The canonical divisor $K_S$ is ample in both cases.
• The fibration structure is determined by the geometry of the bicanonical system and the action of the involution.
• The moduli space of such surfaces is not isolated, despite the expected dimension being zero, indicating a rich deformation theory.