Slope inequality for families of curves over surfaces

Abstract

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 $K_{X/Y}^n/{\mathrm{ch}}_{n-1}(f_{\ast }{\omega }_{X/Y})$. 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 $\dim Y=1$.

One more section is added. Other minor changes are made. Comments are very welcome

References in corpus (2)

• The Oort conjecture on Shimura curves in the Torelli locus of curves
• On positivity and semistability of vector bundles in finite and mixed characteristics