Stringy E-functions of hypersurfaces and of Brieskorn singularities

TL;DR

This paper connects stringy E-functions of hypersurfaces to motivic zeta functions, providing computational methods and analyzing the positivity of stringy Hodge numbers, especially in relation to Brieskorn singularities.

Contribution

It establishes a link between stringy E-functions and motivic zeta functions, and introduces an accessible method to compute contributions from Brieskorn singularities.

Findings

Stringy E-functions can be derived from motivic zeta functions via residues.

The paper provides an algorithm to compute stringy E-functions from Newton polyhedra.

Counterexample in 6 dimensions shows nonnegativity of stringy Hodge numbers does not always hold.

Abstract

We show that for a hypersurface Batyrev's stringy E-function can be seen as a residue of the Hodge zeta function, a specialization of the motivic zeta function of Denef and Loeser. This is a nice application of inversion of adjunction. If an affine hypersurface is given by a polynomial that is non-degenerate with respect to its Newton polyhedron, then the motivic zeta function and thus the stringy E-function can be computed from this Newton polyhedron (by work of Artal, Cassou-Nogues, Luengo and Melle based on an algorithm of Denef and Hoornaert). We use this procedure to obtain an easy way to compute the contribution of a Brieskorn singularity to the stringy E-function. As a corollary, we prove that stringy Hodge numbers of varieties with a certain class of strictly canonical Brieskorn singularities are nonnegative. We conclude by computing an interesting 6-dimensional example. It shows that a result, implying nonnegativity of stringy Hodge numbers in lower dimensional cases, obtained in our previous paper, is not true in higher dimension.