Slope inequality for families of curves over surfaces

TL;DR

This paper introduces a new slope inequality for families of curves over surfaces, providing lower bounds in characteristic zero and positive characteristic, and proposes a generalized notion of slope for higher-dimensional bases.

Contribution

It defines a new slope concept for families of curves over surfaces and proves a lower bound using characteristic p methods, extending known results to higher dimensions.

Findings

Established a slope inequality for $ ext{dim } Y=2$ over any characteristic.

Proved compatibility of the new slope with lower-dimensional cases.

Introduced a novel proof technique using characteristic p geometry.

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