On lower bounds for the F-pure threshold of equigenerated ideals

Abstract: Let $k$ be a field of characteristic $p > 0$ and $R = k[x_0,\dots, x_n]$. We consider ideals $I\subseteq R$ generated by homogeneous polynomials of degree $d$. Takagi and Watanabe proved that $\mathrm{fpt}(I)\geq \mathrm{height}(I)/d$; we classify ideals $I$ for which equality is attained. Additionally, we describe a new relationship between $\mathrm{fpt}(I)$ and $\mathrm{fpt}(I|_H)$, where $H$ is a general hyperplane through the origin. As an application, for degree-$d$ homogeneous polynomials $f\in R$ when $p \geq n-1$, we show that either $p$ divides the denominator of $\mathrm{fpt}(f)$ or $\mathrm{fpt}(f)\geq r/d$, where $r$ is the codimension of the singular locus of $f$.