Product of positive-genus curves is not a projective hypersurface
= Product of positive-genus curves is not a projective hypersurface
If $C$ and $D$ have positive genus, then $q(C\times D)=g(C)+g(D)>0$, whereas every smooth surface hypersurface in projective space lies in $\mathbb P^3$ and has irregularity zero. Hence $C\times D$ is not isomorphic to a projective hypersurface.