Zeta functions of projective hypersurfaces with ordinary double points

Abstract

We extend the approach of 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 \(f \in {\mathbb {Z}}[x_0, \ldots , x_n]\), 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 \({\mathbb {P}}^3\).