Zeta functions of projective hypersurfaces with ordinary double points

Abstract

We extend the approach Abbott, Kedlaya and Roe to computation of the zeta function of a projective hypersurface with $\tau$ isolated ordinary double points over a finite field ${\mathbb{F}}_q$ given by the reduction of a homogeneous polynomial $f\in \mathbb{Z}[x_0,\dots ,x_n]$, under the assumption of equisingularity over ${\mathbb{Z}}_q$. The algorithm is based on the results of Dimca and Saito (over the field $\mathbb{C}$ of complex numbers) on the pole order spectral sequence in the case of ordinary double points. We give some examples of explicit computations for surfaces in ${\mathbb{P}}^3$.

Versions 2: some typos fixed and a reference [DiSt2] added