Surfaces with $p_g = q= 1$, $K^2 = 7$ and non-birational bicanonical mpas

TL;DR

This paper studies minimal surfaces of general type with specific invariants, proving the bicanonical map's degree is 1 or 2 and describing the surface when the degree is 2.

Contribution

It establishes the degree of the bicanonical map for these surfaces and characterizes the surface when the map has degree 2.

Findings

Degree of bicanonical map is 1 or 2

Surfaces with degree 2 are described as double covers

Provides classification for these surfaces

Abstract

Let $S$ be a minimal surface of general type with $p_g = q = 1, K_S^2 = 7$. We prove that the degree of the bicanonical map is 1 or 2. Furthermore, if the degree is 2, we describe $S$ by a double cover.