The College of Science in the Department of Mathematics discussed a seminar entitled Classification of Cubic Surfaces with 27 Lines over the Galois Field of Order 19 for postgraduate student Marwa M. Jawad. In this seminar, we investigate the existence and construction of the twenty-seven lines on a smooth cubic surface, specifically within the geometry of the finite field. Beginning by addressing the classical result of Cayley and Salmon, exploring the algebraic reasons why a non-singular cubic surface must contain exactly twenty-seven lines. The study then shifts to a practical framework within the projective space, detailing the methods used to explicitly find and verify these lines on a given surface. Furthermore, we classify these surfaces based on their configuration of Eckardt points—points where three of the twenty-seven lines concur—providing a clear mapping of the intersection patterns and symmetries inherent to cubic surfaces in a finite geometric setting.
