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Slopes of effective divisors on the moduli space of stable curves. (English) Zbl 0705.14026
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.
Reviewer: A.Papantonopoulou

##### MSC:
 14H10 Families, moduli of curves (algebraic) 14C20 Divisors, linear systems, invertible sheaves
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