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Log canonical models for the moduli space of curves: the first divisorial contraction. (English) Zbl 1172.14018

Let \(\bar M _g\) be the moduli space of stable curves of genus \(g\geq 4\), \(\Delta _i\) be the boundary divisors where \(i=0,\ldots , [g/2]\), \(\bar {\mathcal M }_g\) denotes the moduli stack and \(\delta _i\) its boundary divisors. If \(\Delta =\sum \Delta _i\) and \(\delta =\sum \delta _i\), then for any \(0\leq \alpha \leq 1\), the divisor \(K_{\bar M _g}+\alpha \Delta +\frac{1-\alpha}2\Delta _1\) pulls back to \(K_{\bar {\mathcal M} _g} +\alpha \delta\) and we may identify the log canonical models corresponding to these two divisors. Let \[ \bar M _g(\alpha )=\text{Proj}\left( \bigoplus _{n\geq 0}\Gamma (n(K_{\bar {\mathcal M} _g}+\alpha \delta))\right). \] By results of Mumford, Cornalba and Harris, \(\bar M _g(\alpha )=\bar M _g\) for \(9/11< \alpha \leq 1\).
In the article under review, the authors show that there is a morphism \(\bar M _g\to \bar M _g(9/11)\) which is a divisorial contraction, moreover for any \(7/10< \alpha \leq 9/11\), we have that \(\bar M _g(\alpha )\) is isomorphic to the projective variety \(\bar M ^{ps}_g\) parametrizing pseudostable curves (i.e. connected reduced curves with nodes and cusp sing such that the canonical sheaf is ample and any genus one subcurve meets the rest of the curve in at least two points).

MSC:

14H10 Families, moduli of curves (algebraic)
14E30 Minimal model program (Mori theory, extremal rays)
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