Odd-genus pencils from a four-divisible polarization ($g=1+\frac{1}{2}(m^2{H}^2+m{K}_X\cdot H),4\mid m$)

On a smooth projective surface, fix an ample divisor $H$. For sufficiently large $m$, a general pencil in $|mH|$ is a Lefschetz pencil. The adjunction formula gives the displayed genus. If $m$ is divisible by four, both terms added to $1$ are even integers, so the genus is odd.