Smooth projective hypersurface has zero irregularity

A smooth surface hypersurface $X_d\subseteq {\mathbb{P}}^3$ has $q(X_d)=0$. This follows from the structure-sheaf sequence of a hypersurface and the vanishing of the intermediate cohomology of line bundles on ${\mathbb{P}}^3$.

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.