On Stability and Denominators of F-pure thresholds in Families of Diagonal Hypersurfaces

Abstract: Given a prime number $p$ and a positive integer $m$, we provide a family of diagonal hypersurfaces ${ f_n }<em>{n = 1}<sup>{\infty}$ in $m$ variables, for which the denominator of $\text{ fpt } (f</em>{n})$ (in lowest terms) is always $p$ and whose $F$-pure thresholds stabilize after a certain $n$. We also provide another family of diagonal hypersurfaces ${ g_n }<em>{n = 1}<sup>{\infty}$ in $m$ variables, for which the power of $p$ in the denominator of $\text{ fpt } (g</em>{n})$ (in lowest terms) diverges to $\infty$ as $n \to \infty$. This behavior of the denominator of the $F$-pure thresholds is dependent on the congruence class of $p$ modulo the smallest two exponents of ${ f_n }$ and ${ g_n }$.