The F-pure threshold of quasi-homogeneous polynomials

TL;DR

This paper computes the F-pure threshold for quasi-homogeneous polynomials, including specific cases like curves and Calabi-Yau hypersurfaces, extending previous work in algebraic geometry.

Contribution

It provides explicit calculations of the F-pure threshold for quasi-homogeneous polynomials in various geometric contexts, advancing understanding in positive characteristic algebraic geometry.

Findings

F-pure threshold computed for specific quasi-homogeneous polynomials

Results applied to curves and Calabi-Yau hypersurfaces

Extends previous theoretical work in the field

Abstract

Inspired by the work of Bhatt and Singh (see: arXiv:1307.1171) we compute the $F$-pure threshold of quasi-homogeneous polynomials. We first consider the case of a curve given by a quasi-homogeneous polynomial $f$ in three variables $x,y,z$ of degree equal to the degree of $xyz$ and then we proceed with the general case of a Calabi-Yau hypersurface, i.e. a hypersurface given by a quasi-homogeneous polynomial $f$ in $n+1$ variables $x_0, \ldots, x_n$ of degree equal to the degree of $x_0 \cdots x_n$.