Classification of Non-Singular Cubic Surfaces up to e-invariants

In this thesis, we use the Clebsch map to construct cubic surfaces with twenty-seven lines in PG(3, q) from 6 points in general position in PG(2, q) for q = 17, 19, 23, 29, 31. We classify the cubic surfaces with twenty-seven lines in three dimensions (up to e- invariants) by introducing computational and geometrical procedures for the classi- fication. All elliptic and hyperbolic lines on a non-singular cubic surface in PG(3, q) for q = 17, 19, 23, 29, 31 are calculated. We define an operation on triples of lines on a non-singular cubic surface with 27 lines which help us to determine the exact value of the number of Eckardt point on a cubic surface. Moreover, we discuss the irreducibil- ity of classes of smooth cubic surfaces in PG(19, C), and we give the corresponding codimension of each class as a subvariety of PG(19, C).