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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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