F-pure threshold and height of quasi-homogeneous polynomials

Abstract

We consider a quasi-homogeneous polynomial $f \in \mathbb{Z}[x_0, \ldots, x_N]$ of degree $w$ equal to the degree of $x_0 \cdots x_N$ and show that the $F$-pure threshold of the reduction $f_p \in \mathbb{F}_p[x_0, \ldots, x_N]$ is equal to the log canonical threshold if and only if the height of the Artin-Mazur formal group associated to $H^{N-1}\left( X, {\mathbb{G}}_{m,X} \right)$, where $X$ is the hypersurface given by $f$, is equal to 1. We also prove that a similar result holds for Fermat hypersurfaces of degree $>N+1$. Furthermore, we give examples of weighted Delsarte surfaces which show that other values of the $F$-pure threshold of a quasi-homogeneous polynomial of degree $w$ cannot be characterized by the height.