$F$-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∈ℤ[x0,…,xN] of degree w equal to the degree of x0⋯xN and show that the F-pure threshold of the reduction fp∈𝔽p[x0,…,xN] is equal to the log-canonical threshold of f if and only if the height of the Artin–Mazur formal group associated to HN−1(X,𝔾m,X), 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.