The Cornalba-Harris slope inequality gives a lower bound on the slope of a family of semistable curves of genus \(g\) parameterized by an integral curve, whose general member turns to be a smooth curve. The present paper by the authors concerns with this celebrated inequality.
Let \(\mathrm{CE}(\overline{\mathcal{M}}^1_{g, 4})\), the so called “Casnati-Ekedahl” locus, denotes the closure of the locus in \(\overline{\mathcal{M}}^1_{g, 4}\) corresponding to curves with non balanced bundle of conics. Then, as for their main result in their “Main Theorem”, under some circumstances, they obtain:

(1)

If \(g\) is odd and the modular image of the parameter space doesn’t locate inside \(\mathrm{CE}(\overline{\mathcal{M}}^1_{g, 4})\), then \(s(f)\geq \frac{16(g-1)}{3g+1}\), where \(f: S\rightarrow B\) denotes the map of the family.

(2)

With a further assumption which they call it “condition (†)”, they prove \(s(f)\geq \frac{16(g-1)}{3g+2}\), when the genus is even.

(3)

If \(F\) denotes the general fiber of \(f\); the \(4\)-gonal morphism \(h: F\rightarrow \mathbb{P}^1\) does not factorise and if the condition (†) holds, then they show \(s(f)\geq \frac{24(g-1)}{5g+3}\). They further obtain, in this case, an equivalent condition guaranteeing the occurrence of their sharp bound.

(4)

If the \(4\)-gonal morphism \(h: F\rightarrow \mathbb{P}^1\) factorises through a double cover of a hyperelliptic curve of genus \(\gamma< \frac{g-3}{6}\) and if their condition (†) is satisfied, then they obtain \(s(f)\geq \frac{4(g-1)}{g-\gamma}\).

An sketch of their proof goes as follows.
In section \(2\), for a fibration \(f: S\rightarrow B\), which factors through a finite Gorenstein cover \(\pi:S\rightarrow Y\) of a ruled surface over \(B\), they express the slope of \(f\) in terms of the \(\pi\)-relative canonical divisor and the Chern classes of the reduced direct image sheaf. Under some positivity assumption in Theorem \(2. 10\), they establish some Bogomolov type inequalities between the Chern classes of a rank two vector bundle on a ruled surface.
Among other results in section \(3\), using Theorem \(2.10\) and some additional arguments they conclude their main result when \(S\) is a Gorenstein cover of a ruled surface.
In their last section, the authors define the property of having a good Gorenstein factorization, under which they estimate the invariants of \(f: X\rightarrow B\) in terms of those of a family \(\tilde{f}: \tilde{X} \rightarrow B\), where \(\tilde{X}\) is Gorenstein. By showing that the bound in their main theorem hold for \(s(\tilde{f})\) together with observing that \(S\) is a minimal model of \(\tilde{X}\), they conclude their main result.
The authors announce that under a particular numerical circumstance, their result is consistent with a result of A. Patel in his Ph. D. thesis. They use a theorem of A. Moriwaki (see Theorem 2.2.1 in: [J. Reine Angew. Math. 480, 177–195 (1996; Zbl 0861.14027)]) to show this coincidence.