The F-pure threshold of quasi-homogeneous polynomials
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Abstract
To any polynomial $fin K[x_0, ldots, x_n]$, where $K$ is a field of characteristic $p>0$, one can attach an invariant called the $F$-pure threshold, first defined by Takagi and Watanabe.
This invariant is the characteristic $p$ analogue of the log-canonical threshold in characteristic zero. The $F$-pure threshold, which is a rational number, is a quantitative measure of the severity of the singularity of $f$. Smaller values of the $F$-pure threshold correspond to a worse'' singularity.
Inspired by the work of Bhatt and Singh we compute the $F$-pure threshold of quasi-homogeneous polynomials, i.e. polynomials $f in K[x_0, ldots, x_n]$ which are homogeneous with respect to some $mathbb{N}$-grading of $K[x_0, ldots, x_n]$. In particular, we consider the 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 $x_0 cdots x_n$. Moreover, we relate the $F$-pure threshold of $f in R=K[x_0, ldots, x_n]$ to a numerical invariant of $X=Proj(R/fR)$, namely the order of vanishing of the so-called Hasse invariant on a certain deformation space of $X$.
In the second part of this thesis we turn our attention away from the Hasse in-variant towards an important invariant in the theory of formal groups. Namely, we 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 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 denote by $X$ the hypersurface given by $f=0$. We 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 of $f$ if and only if the height of the Artin-Mazur formal group associated to $H^{n-1}left( X, {mathbb{G}}_{m,X} right)$ 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 quasi-homogeneous polynomial of degree $w$ cannot be characterized by the height.