Odd-genus pencils from a four-divisible polarization
= Odd-genus pencils from a four-divisible polarization
{title2=$g=1+\tfrac12(m^2H^2+mK_X\cdot H),\quad4\mid m$}
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.