Application of Algebraic Geometry In Three Dimensional projective space PG (3,7)

Abstract

The main goal of this work is to construct surfaces and complete arcs in the projective 3 – space PG (3, q) over Galois fields GF (p), p=7. Which represents applications of algebraic geometry in three-dimensional projective space PG (3, P), where p=7 which is a (k, ƪ)-span. We get the following results. First, we found the points, lines and planes in PG (3,7) and we construct (k, ƪ)-span which is a set of k lines no two of which intersect. We prove that the maximum complete (k, ƪ)-span in PG (3,7) is (50, ƪ)-span, which is the equal to all the points of the space that is called a spread. Second in general we prove geometrical rule the total number of Spread in projective space PG (3, p) where p is prime, P ≥ 2 is p 2 + l.