Slope inequality for families of curves over surfaces

In this paper, we investigate the general notion of the slope for families of curves f: X→ Y. The main result is an answer to the above question when dim Y= 2 , and we prove a lower bound for this new slope in this case over fields of any characteristic. Both the notion and the slope inequality are compatible with the theory for dim Y= 0 , 1 in a very natural way, and this gives a strong evidence that the slope for an n-fold fibration of curves f: X→ Y may be KX/Yn/chn-1(f∗ωX/Y). Rather than the usual stability methods, the whole proof of the slope inequality here is based on a completely new method using characteristic p> 0 geometry. A simpler version of this method yields a new proof of the slope inequality when dim Y= 1.