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→Yf:X→Y . The main result is an answer to the above question when dimY=2dim⁡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 dimY=0,1dim⁡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→Yf:X→Y may be KnX/Y/chn−1(f∗ωX/Y)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>0p>0 geometry. A simpler version of this method yields a new proof of the slope inequality when dimY=1dim⁡Y=1