 Solution

True for a smooth complex projective surface, the usual manifold convention in this question. Choose an ample divisor $H$. The auxiliary pencil-existence theorem states that, for all sufficiently large $m$, a general two-dimensional space of sections of ${\mathcal{O}}_X(mH)$ defines a Lefschetz pencil: its base points are transverse and finite, its general fibre is a smooth connected curve, and each singular fibre has exactly one ordinary node, with distinct critical values. High powers separate the required jets, so a generic pencil has precisely these singularities.

Choose such an $m$ divisible by four, and write $D=mH$. The adjunction formula for a smooth connected curve of class $D$ on a smooth projective surface is $2g-2=D\cdot (D+K_X)$. ConsequentlyWriting $m=4\ell$ gives $g=1+8{\ell }^2{H}^2+2\ell K_X\cdot H$, an odd integer. Thus a sufficiently positive four-divisible polarization gives an odd-genus Lefschetz pencil. The surface is taken to be smooth; a singular projective variety need not support a Lefschetz pencil in this smooth-manifold sense.