Log minimal model program for the moduli space of stable curves: the first flip. (English) Zbl 1273.14034

Let \(\bar M _g\) denote the moduli space of stable curves of genus \(g\). This space is known to be of general type for all \(g\geq 24\) and so it is birational to its canonical model which is given by \(\text{Proj}\bigoplus _{n\geq 0}\Gamma (\bar M _g,nK_{\bar{\mathcal M}_g})\) (note that this ring is finitely generated by C. Birkar et al., [J. Am. Math. Soc. 23, No. 2, 405–468 (2010; Zbl 1210.14019)]). It is an important natural problem to understand the geometry of this canonical model. One possible strategy is to approximate the canonical model by log canonical models. Let \(\delta\) be the boundary divisor. By B. Hassett and D. Hyeon [Trans. Am. Math. Soc. 361, No. 8, 4471–4489 (2009; Zbl 1172.14018)] it is known that \( K_{\bar{\mathcal M}_g}+\alpha \delta\) is ample for \(9/11<\alpha \leq 1\) (and hence \(\bar{\mathcal M}_g\cong \bar{\mathcal M}_g(\alpha ):=\text{Proj}\bigoplus _{n\geq 0}\Gamma (\bar M _g,n(K_{\bar{\mathcal M}_g}+\alpha \delta ))\)) there is a divisorial contraction \( \bar{\mathcal M}_g\to \bar{\mathcal M}_g(9/11)\) and \( \bar{\mathcal M}_g(\alpha) \cong \bar{\mathcal M}_g(9/11)\) for \(7/10<\alpha \leq 9/11\). In this paper it is shown that the map the morphism \( \bar{\mathcal M}_g(\frac 7{10}+\epsilon )\to \bar{\mathcal M}_g(\frac 7{10})\) is a flipping contraction and the rational map \( \bar{\mathcal M}_g(\frac 7{10}+\epsilon )\dasharrow \bar{\mathcal M}_g (\frac 7{10}-\epsilon )\) is a flip for \(0<\epsilon \ll 1\). The geometry of this flip is explicitely described in terms of invariant theory (GIT).


14E30 Minimal model program (Mori theory, extremal rays)
14H10 Families, moduli of curves (algebraic)
14L24 Geometric invariant theory
Full Text: DOI arXiv


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