$F$-pure threshold and height of quasihomogeneous polynomials

Abstract

The aim of this paper is to give a connection between the $F$-pure threshold of a polynomial and the height of the corresponding Artin–Mazur formal group. For this, we consider a quasihomogeneous polynomial $f\in \mathbb{Z}\left[x_0,\dots ,x_N\right]$ of degree $w$ equal to the degree of $x_0\cdots x_N$ and show that the $F$-pure threshold of the reduction $f_p\in {\mathbb{𝔽}}_p\left[x_0,\dots ,x_N\right]$ is equal to the log-canonical threshold of $f$ 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 greater than $N+1$. Furthermore, we give examples of weighted Delsarte surfaces which show that other values of the $F$-pure threshold of a quasihomogeneous polynomial of degree $w$ cannot be characterized by the height.