Jet schemes of quasi-homogeneous hypersurfaces and motivic monodromy conjecture for isolated quasi-homogeneous hypersurface singularities

Abstract

Let $k$ be a field. Let $m\in\mathbf{N}_{>0}$ be a positive integer. Let $f\in k[x_1,\ldots,x_m]$ be a polynomial with degree $d\geq 1$ and associated hypersurface $H:=H(f):=\mathrm{Spec}(k[x_1,\ldots,x_m]/\langle f\rangle)$. In this article, we firstly provide a structure property of the weighted-homogeneity of $f$ in terms of the jet schemes ${\mathscr{L}_{H}}$ of $H$. As a by-product, we deduce from this property a new and very \linebreak effective method for the computation of the motivic Poincaré power series $P_H(T):=\sum_{n\geq 0} [\mathscr{L}_{n}(H)]T^n\in K_0(\mathrm{Var}_k)[[T]]$ associated with a homogeneous hypersurface $H$ with a single isolated singularity at the origin $\frak o$ (and more generally with a specific class of isolated quasi-homogeneous hypersurface singularities). With this point of view we obtain various consequences. For the considered class of varieties, our method provides a characteristic-free proof of the rationality of $P_H(T)$ which does not use motivic integration nor the existence of resolutions of singularities; we obtain a precise description of the numerator and the possible poles in the rational expression of $P_H(T)$; when the field is assumed to be of characteristic zero, this allows us to prove the validity of the motivic monodromy conjecture.