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 \ fₙ = 1∞ in m variables, for which the denominator of fpt (fₙ) (in lowest terms) is always p and whose F-pure thresholds stabilize after a certain n. We also provide another family of diagonal hypersurfaces \ gₙ = 1∞ in m variables, for which the power of p in the denominator of fpt (gₙ) (in lowest terms) diverges to ∞ as n → ∞. This behavior of the denominator of the F-pure thresholds is dependent on the congruence class of p modulo the smallest two exponents of \ fₙ \ and \ gₙ \.