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

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.