Remarks on surfaces with c12=2χ-1 having non-trivial 2-torsion

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.