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.