  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 $f:
X \to Y$ may be $K_{X/Y}^n / \mathrm{ch}_{n-1}(f_* \omega_{X/Y})$.
  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 $\dim Y = 1$.
