Decomposition of the Diagonal and Eigenvalues of Frobenius for Fano Hypersurfaces

Let X ⊂ P n be a possibly singular hypersurface of degree d ≤ n, defined over a finite field F q. We show that the diagonal, suitably interpreted, is decomposable. This gives a proof that the eigenvalues of the Frobenius action on its ℓ-adic cohomol-ogy H i (¯ X, Q ℓ), for ℓ = char(F q), are divisible by q, without using the result on the existence of… CONTINUE READING