Decomposition of the diagonal and eigenvalues of Frobenius for Fano hypersurfaces

Let $X\subset {\mathrm{\P }}^n$ be a possibly singular hypersurface of degree $d\le n$, defined over a finite field $k$. We show that the diagonal, suitably interpreted, is decomposable. This gives a proof that the eigenvalues of the Frobenius action on its $\mathrm{\ell }$-adic cohomology $H^i(\bar{X},{\text{Q}}_{\mathrm{\ell }})$, for $\mathrm{\ell }\mathrm{\ne }\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}(k)$, are divisible by $q$, without using the result on the existence of rational points by Ax and Katz.