We shall show that any complex minimal surface of general type with $c_1^2 = 2\chi -1$ having non-trivial 2-torsion divisors, where $c_1^2$ and $\chi$ are the first Chern number of a surface and the Euler characteristic of the structure sheaf respectively, has the Euler characteristic $\chi$ not exceeding 4. Moreover, we shall give a complete description for the surfaces of the case $\chi =4$, and prove that the coarse moduli space for surfaces of this case is a unirational variety of dimension 29. Using the description, we shall also prove that our surfaces of the case $\chi = 4$ have non-birational bicanonical maps and no pencil of curves of genus 2, hence being of so called non-standard case for the non-birationality of the bicanonical maps.
Masaaki MURAKAMI. "Remarks on surfaces with $c_1^2 =2\chi -1$ having non-trivial 2-torsion." J. Math. Soc. Japan 65 (1) 51 - 95, January, 2013. https://doi.org/10.2969/jmsj/06510051