Portugaliae Mathematica

Let S be a surface of general type with not birational bicanonical map
\nand that does not contain a pencil of genus 2 curves. If K^2_S= 8, p_g(S) = 4 and q(S) = 0
\nthen S can be given as double cover of a quadric surface. We show that its moduli space
\nis generically smooth of dimension 38, and single out an open subset. Note that for these
\nsurfaces h2(S; TS) is not zero.