A Note on k-Secant Lines of the Klein Cubic Threefold in PG(4,3)

In this paper, we developed an algorithm to classify and construct k-secant lines for the Klein cubic threefold F in the projective space PG(4,3). The algorithm identifies secant lines based on their intersections with F, revealing distinct categories of non-secant and k-secant lines. Additionally, we partitioned the point set of F into three subsets, uncovering geometric configurations and forming 5-gons related to perspectively. This analysis offers new insights into the structure of the Klein cubic threefold and its geometric properties.