Definitions and Introduction

This is a continuation of the papers of Braun and Fløystad [BF], Cook [C1] and Braun and Cook [BC] bounding the degree of smooth surfaces not of general type in P using Generic Initial Ideal Theory. We will use and expand on the results of [BC] using Liaison Theory. We will further restrict the configurations of the generic initial ideal of a generic hyperplane section of the surface by considering the geometric implications of having between one and five generators of the generic initial ideal in degree much higher than the other generators. We prove the following