Maximal number of skew lines on Fermat surfaces

It is well known that the Fermat surface of degree d≥ 3 has 3d² lines. However, it has not yet been established what is the maximal number of pairwise disjoint lines that it can have if d≥ 4 . In this article, we show that the maximal number of skew lines on the Fermat surface of degree d≥ 4 is 3d , either d even or d odd distinct of 5, otherwise ( d=5 ) it contains no more than 13 pairwise disjoint lines.