Finding Rational Solutions on a Nonsingular Cubic Surface in P^3

The purpose of this thesis is to present a strategy for parametrizing the rational points on nonsingular, homogeneous cubic surfaces. The particular Diophantine equation we will consider is X3 + Y 3 + Y Z2 + W 3 = 0. The strategy will be to find a family of singular cubic curves on our hypersurface with which to sweep through rational solutions, not unlike the standard parametrization of the circle. We enumerate the 27 lines on the surface in order to search for a rational point. The tangent plane is known to intersect a nonsingular cubic surface in a singular cubic curve. We present a 2-parameter set of solutions and discuss its possible incompleteness. Throughout, we provide an introduction to the necessary algebraic geometry, as well as presenting, in detail, our parametrization. We end with an explanation of how our process can be generalized for similar equations vis-a-vis a computer algebra package, such as SAGE.