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

Given a prime number $p$ and a positive integer $m$, we provide a family of diagonal hypersurfaces $\{f_n{\}}_{n=1}^{\mathrm{\infty }}$ in $m$ variables, for which the denominator of $\text{ fpt }(f_n)$ (in lowest terms) is always $p$ and whose $F$-pure thresholds stabilize after a certain $n$. We also provide another family of diagonal hypersurfaces $\{g_n{\}}_{n=1}^{\mathrm{\infty }}$ in $m$ variables, for which the power of $p$ in the denominator of $\text{ fpt }(g_n)$ (in lowest terms) diverges to $\mathrm{\infty }$ as $n\to \mathrm{\infty }$. 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 $\{g_n\}$.