Decomposition of the diagonal and eigenvalues of Frobenius for Fano hypersurfaces

Let X ⊂ [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /] be a possibly singular hypersurface of degree d ≤ n, defined over a finite field [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /]. We show that the diagonal, suitably interpreted, is decomposable. This gives a proof that the eigenvalues of the Frobenius action on its ℓ-adic cohomology Hⁱ([inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="03i" /]), for ℓ ≠ char([inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="04i" /]), are divisible by q, without using the result on the existence of rational points by Ax and Katz.