[Paper Review] Involutions on a surface of general type with $p_g=q=0$, $K^2=7$

Yongnam Lee, YongJoo Shin|Osaka City University (Osaka City University)|Mar 18, 2010

Algebraic Geometry and Number Theory12 references4 citations

TL;DR

This paper classifies the birational models and branch divisors of quotient surfaces arising from involutions on minimal surfaces of general type with $p_g = q = 0$ and $K^2 = 7$. Using blow-ups and resolution of singularities, it proves that when the quotient is birational to an Enriques surface, the involution has exactly 9 fixed points and the branch divisor $B_0$ takes specific configurations involving curves of genus 1, 3, or 4 with given self-intersection numbers.

ABSTRACT

In this paper we study on the involution on minimal surfaces of general type with $p_g=q=0$ and $K^2=7$. We focus on the classification of the birational models of the quotient surfaces and their branch divisors induced by an involution.

Motivation & Objective

• To classify the birational models of quotient surfaces and their branch divisors induced by involutions on minimal surfaces of general type with $p_g = q = 0$ and $K^2 = 7$.
• To determine whether such surfaces can have quotients birational to an Enriques surface, building on known examples for $K^2 = 1,2,3,4$.
• To exclude the possibility of such quotients for $K^2 = 8,9$ and focus on the open cases $K^2 = 5,6,7$.
• To identify the precise configurations of the branch divisor $B_0$ in the quotient surface model, particularly when the quotient is birational to an Enriques surface.
• To establish that such surfaces possess a 2-torsion element and have exactly 9 fixed points under the involution.

Proposed method

• Construct a commutative diagram involving the minimal surface $S$, its blow-up $V$ at $k$ fixed points, the quotient $W = S/\sigma$, and the minimal resolution $\Sigma$ of the quotient singularities.
• Use the blow-up $\epsilon$ of $S$ at $k$ isolated fixed points to resolve the quotient singularities, leading to the surface $V$ and the induced map $\tilde{\pi}: V \to W$.
• Analyze the branch divisor $B_0 = \tilde{\pi}(R_0)$, where $R_0 = \epsilon^*(R)$ and $R$ is the fixed curve of the involution $\sigma$ on $S$, using adjunction and Hurwitz formulas.
• Compute the canonical degree $K_W^2$ and the self-intersection and geometric genus of irreducible components $\Gamma_i$ of $B_0$ via intersection theory on the intermediate surfaces $T_i$ and $\Sigma_i$, tracking $(-1)$-curves and nodes.
• Apply results from [6] and [8] to determine whether $W$ is rational, properly elliptic, of general type, or birational to an Enriques surface based on $K_W^2$ and $H^0(W, \mathcal{O}_W(2K_W))$.
• Use the classification of $k$-fixed point cases ($k = 5,7,9,11$) to systematically enumerate possible $B_0$ configurations and their geometric invariants.

Experimental results

Research questions

• RQ1 What are the possible configurations of the branch divisor $B_0$ on the quotient surface $W$ when an involution acts on a minimal surface of general type with $p_g = q = 0$ and $K^2 = 7$?
• RQ2 Can such a quotient surface $W$ be birational to an Enriques surface, and if so, what are the necessary conditions on the fixed locus and $B_0$?
• RQ3 How does the number of fixed points $k$ of the involution relate to the canonical degree $K_W^2$ and the birational type of $W$?
• RQ4 What are the geometric invariants (genus, self-intersection) of the irreducible components of $B_0$ in the case where $W$ is birational to an Enriques surface?
• RQ5 Does such a surface necessarily admit a 2-torsion element in its Picard group, and how does this relate to the structure of the involution?

Key findings

• When the quotient $W = S/\sigma$ is birational to an Enriques surface, the involution $\sigma$ has exactly 9 fixed points, and the fixed divisor $R$ on $S$ is either a curve of genus 3 or a union of a genus 1 and a genus 3 curve.
• The branch divisor $B_0$ on the minimal model of $W$ takes one of the following forms: $\begin{subarray}{c}\Gamma_0\\(3,0)\end{subarray} + \begin{subarray}{c}\Gamma_1\\(1,-2)\end{subarray}$, $\begin{subarray}{c}\Gamma_0\\(3,-2)\end{subarray}$, or other configurations with $\Gamma_0$ of genus 3 or 4 and $\Gamma_1$ of genus 1 or 2.
• For $k = 9$, $K_W^2 = -2$, and $W$ is birational to an Enriques surface if and only if $B_0$ is $\begin{subarray}{c}\Gamma_0\\(3,0)\end{subarray} + \begin{subarray}{c}\Gamma_1\\(1,-2)\end{subarray}$ or $\begin{subarray}{c}\Gamma_0\\(3,-2)\end{subarray}$.
• When $k = 11$, the bicanonical map of $S$ is composed with the involution, and $W$ is rational, with $K_W^2 = -4$, and $B_0$ not fully classified here due to prior work.
• The surface $S$ admits a 2-torsion element in its Picard group, as shown by the structure of the involution and the quotient geometry.
• The classification shows that $W$ is of general type, properly elliptic, or rational depending on $k$ and $K_W^2$, with $k = 5$ giving $K_W^2 = 2$ and $B_0 = \begin{subarray}{c}\Gamma_0\\(1,-2)\end{subarray}$, and $k = 7$ giving $K_W^2 = 1$ or $0$ with various $B_0$ types.