The Classification Of Surfaces With Clifford Index 2

We are familiar to Clifford's Theorem, by which we can define the Clifford index of curves. They paly an important role in the theory of algebraic curves. Many important and beautiful results are induced from this theorem. How to generalize Clifford's Theorem on surfaces is an interesting problem in algebraic geometry.Hao sun has given a result similar to Clifford's Theorem and de-fined the Clifford Index of surfacesα(X). Moreover, he has classified the surfaces with Clifford Index 0 and 1.In this paper we want to give out the classification of surfaces with Clifford Index 2. Our main method is to take advantage of the properties of linear system and the theory of double covering. We systematically analyze the mapping induced by linear system with minimal Clifford Index.In the case of pencils, we prove the properties of the fibration induced by the pencil with minimal Clifford Index:the genus of the curve is no more than 1 and the genus of fibres is no more 3. Especially, when the curve is elliptic curves, the fibre is of genus 2.In the other cases, we prove that the mapping induced by the linear system must be a double covering onto a surface with mini-mal degree. Moreover, the linear system has no fixed point and the self-intersection number of its divisor is 2. With the classical clas-sification of surfaces with minimal degree[23], we discuss the cases one by one. and prove that the image of the mapping must be P2. Besides, we get the bounds of the invariants:11≤Kx2≤18, 2≤x((?)x)≤10.