Slope inequality for families of curves over surfaces

In this paper, we investigate the general notion of the slope for families of curves $f:X\to Y$. 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 $f: X \to Y$maybe$K_{X/Y}^n / $\mathrm{c}\mathrm{h}${n-1}(f* \omega_{X/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 $\dim Y=1$.