Zeta functions of projective hypersurfaces with ordinary double points

TL;DR

This paper develops an algorithm to compute the zeta function of projective hypersurfaces with ordinary double points over finite fields, extending previous methods and providing explicit examples for surfaces in projective 3-space.

Contribution

It introduces a novel algorithm based on pole order spectral sequences for hypersurfaces with ordinary double points, extending prior work to a broader class of singularities.

Findings

Algorithm successfully computes zeta functions for specific hypersurfaces.

Explicit examples demonstrate practical application of the method.

Extension of existing techniques to singular 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 $f \in \mathbb{Z}[x_0, \ldots, x_n]$, 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 $\mathbb{P}^3$.