The Classification Of Algebraic Surface Of General Type With Small Volume In Positive Characteristic

The classification of algebraic varieties is an important topic in algebraic geometry.The classification of surfaces whose Kodaira dimension are-∞,0,1 are quite clear.But we still know little about the classification of surfaces whose Kodaira dimension are 2.This kind of surfaces is called the surface of general type.In 1875,Max Noether proved the Noether inequality that K_X~2≥2p_g(X)-4 for a minimal surface X of general type.This inequality shows the relation between the intersection number of the canonical bundle of the surface and the geometric genus of the surface.When the equality holds,we say that X is on the Noether line.In 1970s,Horikawa finished the classification of the minimal surface of general type over complex numbers which is near the Noether line.In this paper,we try to study the topic in positive characteristic.The main problem is that in positive characteristic some analytic properties no longer hold.We will establish some new method to solve this problem.