The \(F\)-pure threshold of quasi-homogeneous polynomials (Q2401909)

Let \(R=K[x_0,\ldots,x_n]\), where \(K\) is a field of characteristic \(p>0\), and \(f\) be a polynomial in \(R\). The F-pure threshold \(\text{fpt}(f)\) of \(f\) is a numerical invariant that measures how severe the singularities of \(f\) are, and it has several important relations with the log-canonical threshold in equal characteristic zero. In [Math. Ann. 362, No. 1--2, 551--567 (2015; Zbl 1328.13006)], \textit{B. Bhatt} and \textit{A. K. Singh} prove explicit formulas for the F-pure threshold of homogeneous polynomials (with respect to the standard grading), that define Calabi-Yao hypersurfaces. In the article under review, the author extends their work to the case when \(f\) is quasi-homogeneous, that is, homogeneous with respect to some weight assigned to the variables. When \(n=2\) and \(C=\text{Proj}(R/fR)\) is an elliptic curve, the author shows that \(\text{fpt}(f)\) equals \(1\) if \(C\) is ordinary, and it equals \(1- \frac{1}{p}\) otherwise. More generally, under some assumptions on the characteristic, the author shows that \(\text{fpt}(f) = 1 - \frac{h}{p}\), where \(h\) is the order of vanishing of the Hasse invariant at the point corresponding to the Calabi-Yau hypersurface \(\text{Proj}(R/fR)\) in some weighted \(n\)-dimensional projective space. The connection with the Hasse invariant is done by looking at injectivity of maps induced by Frobenius on graded components of certain local cohomology modules.