On the Singular Fibers of a Fibration Structure on Fermat Surfaces

Let \(k\) be an algebraically closed field with positive characteristic \(p\) and let \(S\) be a Fermat surface of degree \(n\) over \(k\). We conduct a study on a fibration \(f: S \to \mathbb{P}^1_k\) and present the relationship between the smoothness of the general fiber of \(f\) and the characteristic \(p\).