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Effective divisors on $$\overline{\mathcal M}_g$$, curves on $$K3$$ surfaces, and the slope conjecture. (English) Zbl 1081.14038
Let $${\mathcal M}_{g}$$ be the moduli space of smooth curves of genus $$g$$, and let $$\overline{\mathcal M}_{g}$$ be the Deligne-Mumford compactification of $${\mathcal M}_{g}$$. The slope $$s(D)$$ of an effective divisor $$D$$ in $$\overline{\mathcal M}_{g}$$ is defined as the smallest rational number $$a/b \geq 0$$ such that the divisor $$a\lambda-b(\delta_{0}+\delta_{1}+\dots \delta_{[g/2]})-[D]$$ is an effective combination of the boundary divisor classes $$\delta_{0}, \delta_{1}, \dots ,\delta_{[g/2]}$$; here $$\lambda$$ is the class of the Hodge line bundle. The slope $$s_{g}$$ of the moduli space $$\overline{\mathcal M}_{g}$$ is defined as $$s_{g}:=\inf \{s(D): D\in \text{Eff}(\overline {\mathcal M}_{g})\}$$.
The slope conjecture of Harris and Morrison predicts that $$s_{g}\geq 6+12/g+1$$. The slope conjecture was shown to be true for all $$g\leq 12$$, $$g\neq 10$$ by J. Harris and I. Morrison [Invent. Math. 99, 321–355 (1990; Zbl 0705.14026)] and S.-L. Tan [Int. J. Math. 9, 119–127 (1998; Zbl 0930.14017)].
The aim of the paper under review is to prove two statements: first that the Harris-Morrison slope conjecture fails to hold on $$\overline{\mathcal M}_{10}$$ and second, that in order to compute the slope of $$\overline{\mathcal M}_{g}$$ for $$g \leq 23$$, one only has to look at the coefficients of the classes $$\lambda$$ and $$\delta_{0}$$ in the standard expansion in terms of the generators of the Picard group. The authors use the fact that the condition that a smooth curve of genus $$g$$ lies on a $$K3$$ surface is divisorial (only) for $$g=10$$, therefore one obtains an effective divisor $$K$$ on the moduli space $${\mathcal M}_{10}$$. Then the authors compute the class of the closure $$\overline{K}$$ of $$K$$ in $$\text{Pic}(\overline{\mathcal M}_{10})$$, obtaining the following formula: $[\overline{K}]=7\lambda-\delta_{0}-5\delta_{1}-9\delta_{2}-12\delta_{3}-14\delta_{4}-B_{5}\delta_{5},$ with $$B_{5}\geq 6$$. The first two coefficients in the above formula were computed by F. Cukierman and D. Ulmer [Compos. Math. 89, No. 1, 81–90 (1993; Zbl 0810.14012)]. From the formula one computes $$s(\overline{K})=7$$, which is strictly smaller than the bound $$78/11$$ predicted by the slope conjecture, so $$\overline{K}$$ gives the promised counterexample.
To compute the class of $$\overline{K}$$ the authors show that $$K$$ has four incarnations as a geometric subvariety of the moduli space $${\mathcal M}_{10}$$. In particular, $$K$$ can be thought of as either (1) the locus of curves $$[C] \in {\mathcal M}_{10}$$ for which the rank 2 Mukai type Brill-Noether locus $$\{E \in \text{SU}_{2}(C, K_{C}): h^{0}(C,E)\geq 7\}$$ is non empty, or (2) the locus of curves $$[C] \in {\mathcal M}_{10}$$ with a non-surjective Wahl map $$\Psi_{K_{C}}: \Lambda^{2}H^{0}(C,K_{C}) \to H^{0}(C,3K_{C})$$ (this second characterization is due to F. Cukierman and D. Ulmer). Note that these characterizations of $$K$$, unlike the original definition of $$K$$, can be extended to other genera $$g \geq 13$$. Finally, the authors give a counterexample to a hypothesis formulated by Harris and Morrison that the Brill-Noether divisors are essentially the only effective divisors on the moduli space of curves having minimal slope $$6+12/g+1$$.

##### MSC:
 14H10 Families, moduli of curves (algebraic) 14H51 Special divisors on curves (gonality, Brill-Noether theory) 14J28 $$K3$$ surfaces and Enriques surfaces 14D20 Algebraic moduli problems, moduli of vector bundles 14C22 Picard groups
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