Decomposition of the diagonal and eigenvalues of Frobenius for Fano hypersurfaces

Details Decomposition of the diagonal and eigenvalues of Frobenius for Fano hypersurfaces Decomposition of the diagonal and eigenvalues of Frobenius for Fano hypersurfaces

2003-02-10

arxiv.org/abs/math/0302109v1

Mathematics

Algebraic Geometry

16 pages

Scientific paper

Let $X\subset \P^n$ 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 $\ell$-adic cohomology $H^i(\bar{X}, \Q_\ell)$, for $\ell \neq {\rm char}(k)$, are divisible by $q$, without using the result on the existence of rational points by Ax and Katz.

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