Let \(\bar {\mathcal M}_ g\) be the moduli space of stable curves of genus g and E the cone of effective divisor classes in \(P=Pic(\bar {\mathcal M}_ g)\otimes {\mathbb{R}}\); let \(\Delta\) be the locus of singular curves in \(\bar {\mathcal M}_ g\), \(\delta\) its class in P, and \(\delta_ i\) the classes in P corresponding to the irreducible components of \(\Delta\). P is generated by the boundary classes \(\delta_ i\) and by \(\lambda\), the class of the Hodge line bundle. The authors describe the intersection of E with the plane spanned by \(\lambda\) and any effective sum \(\gamma\) of the boundary classes, in particular the slopes of the effective cone \(s_{\gamma}\). For the most important of these slopes \(s_ g:=s_{\delta}\) they conjecture that \(s_ g\geq 6+12/(g+1)\) with equality when \(g+1\) is composite. In their main theorem they prove their conjecture for \(2\leq g\leq 5\) and in particular they show that the inequality can be strict if \(g+1\) is prime. They also give a geometric description of effective irreducible divisors with slope \(s_ g\) for \(g=3\) and 5.
The paper contains a very well written introduction on the subject, a description of some consequences of the conjecture and a discussion on why the construction proves the conjecture for small g only.