GIT constructions of log canonical models of \(\overline Mg\) (Q2895542)

Let \(C \subset {\mathbb P}^N\) be a projective curve and let \(m \geq 2\) be an integer such that \(\text{H}^0({\mathbb P}^N, {\mathcal O}_{{\mathbb P}^N}(l)) \rightarrow \text{H}^0(C, {\mathcal O}_C(l))\) is surjective for \(l \geq m - 1\) and \(\text{H}^1(C, {\mathcal O}_C(l)) = 0\) for \(l \geq m\). Let \(P(l) = \chi(C, {\mathcal O}_C(l))\) be the Hilbert polynomial of \(C\). One says that \(C\) is \(m\)-\textit{Hilbert stable} if the point \([C]_m\) of \({\mathbb P}\left(\overset{P(m)}\bigwedge \text{H}^0({\mathbb P}^N, {\mathcal O}_{{\mathbb P}^N}(m))\right)\) deduced from the surjection \(\text{H}^0({\mathbb P}^N, {\mathcal O}_{{\mathbb P}^N}(m)) \rightarrow \text{H}^0(C, {\mathcal O}_C(m))\) is stable with respect to the action of \(\text{SL}(N + 1)\). If this happens for any \(m >> 0\), one says that \(C\) is \textit{asymptotically Hilbert stable}.NEWLINENEWLINED. Giesecker's GIT construction of the compactification \(\overline{M}_g\) of the moduli space of genus \(g \geq 2\) curves is based on proving that the curves whose \(n\)-canonical models, \(n \geq 5\), are asymptotically Hilbert stable are exactly the Deligne-Mumford stable curves (as it is well-known, these curves have only nodes as singularities and each nonsingular rational component intersects the rest of the curve in at least three points). The curves whose 3-canonical models are asymptotically Hilbert stable are exactly the \textit{pseudostable} curves of \textit{D. Schubert} [Compos. Math. 78, No. 3, 297--313 (1991; Zbl 0735.14022)] (they can have ordinary cusps but no elliptic tail is allowed), and the same is true for 4-canonical models, as shown by \textit{D. Hyeon} and \textit{I. Morrison} [Math. Res. Lett. 17, No. 4, 721--729 (2010; Zbl 1271.14065)]. Moreover, the curves of genus \(g \geq 4\) whose bicanonical models are aymptotically Hilbert stable were identified by \textit{B. Hassett} and \textit{D. Hyeon} [Ann. Math. (2) 177, No. 3, 911--968 (2013; Zbl 1273.14034)]; they were called \textit{h-semistable} and can have nodes, cusps and tacnodes but certain chains of elliptic curves are excluded.NEWLINENEWLINEOn the other hand, the study, for small \(m\), of the \(m\)-Hilbert stability of bicanonical (resp., tricanonical if \(g = 2\)) models of curves can be, conjecturally, used to accomplish the log minimal model program for \(\overline{M}_g\) (known also as the ``Hassett-Keel program''). This program aims to realize the various log canonical models: NEWLINE\[NEWLINE \overline{\mathcal M}_g(\alpha) = \text{Proj}\left({ \bigoplus_{l\geq 0} \Gamma(\overline{\mathcal M}_g, l(K_{\overline{\mathcal M}_g} + \alpha \delta))} \right) \, , NEWLINE\]NEWLINE for certain values of \(\alpha \in [0,1]\), as moduli spaces.NEWLINENEWLINEIn the paper under review, the authors treat several examples, illustrating various technical aspects occuring in the above mentioned study. Firstly, they show, using a method developed by \textit{I. Morrison} and \textit{D. Swinarski} [Exp. Math. 20, No. 1, 34--56 (2011; Zbl 1267.14059)] based on a result of \textit{G. R. Kempf} [Ann. Math. (2) 108, No. 2, 299--316 (1978; Zbl 0406.14031)], that the curve \(C \subset {\mathbb P}^4\) parametrized by NEWLINE\[NEWLINE \nu \, : \, {\mathbb P}^1 \longrightarrow {\mathbb P}^4,\;[s,t] \mapsto [s^6, s^4t^2, s^3t^3, s^2t^4, t^6] NEWLINE\]NEWLINE is 2-Hilbert semistable. \(C\) is a tricanonical genus 2 curve with two ordinary cusps and played an important role in the construction of \(\overline{M}_2^{ps}\) by \textit{D. Hyeon} and \textit{Y. Lee} [Math. Ann. 337, No. 2, 479--488 (2007; Zbl 1111.14017)].NEWLINENEWLINEThen, they show that a curve \(C \subset {\mathbb P}^N\) contained in the \(r\)th thickening of a hyperplane \(H \subset {\mathbb P}^N\) is \(m\)-Hilbert unstable for \(m > (N+1)(r-1)\).NEWLINENEWLINENext, the authors illustrate a method developed in the above mentioned papers of D. Hyeon et al., based on the notion of \textit{basin of attraction} of a point fixed by the action of a one parameter subgroup, by proving that if \(C = D \cup_p R\) is a bicanonical genus \(g\) curve, with \(R\) a curve of genus 2 meeting \(D\) in a node \(p\) such that \(p\) is a Weierstrass of \(R\), then \(C\) is \(m\)-Hilbert unstable for \(m \leq 5\) and at best strictly semistable for \(m = 6\). Notice that \(C\) is Deligne-Mumford stable.NEWLINENEWLINEFinally, the authors take a closer look at the flip: NEWLINE\[NEWLINE \overline{\mathcal M}_g\left(\frac{7}{10}+\varepsilon \right) \overset{\Psi}\longrightarrow \overline{\mathcal M}_g\left(\frac{7}{10}\right) \overset{\Psi^+}\longleftarrow \overline{\mathcal M}_g\left(\frac{7}{10}-\varepsilon \right) NEWLINE\]NEWLINE defined in the above mentioned paper of Hassett and Hyeon. Hassett and Hyeon showed that \(\overline{\mathcal M}_g\left(\frac{7}{10}+\varepsilon \right)\) can be identified with Schubert's 3-canonical quotient \(\overline{M}_g^{ps}\), that \(\overline{\mathcal M}_g\left(\frac{7}{10}-\varepsilon \right)\) can be identified with \(\overline{M}_g^{hs}\), and that \(\overline{\mathcal M}_g\left(\frac{7}{10}\right)\) is the quotient of the Chow variety of Chow-stable bicanonical curves. Using the results of \textit{J. Alper, D. I. Smyth} and \textit{F. van der Wyck} [``Weakly proper moduli stacks of curves'', e-print, \url{arXiv:1012.0538} (2010)], the authors of the paper under review analyse the above flip locally (in the étale topology) arround the point \(D \cup R_1 \cup R_2\) of \(\overline{M}_g^{cs}\), where \(D\) is a smooth curve of genus \(g - 2\), and \(R_1\), \(R_2\) are rational curves meeting each other in a tacnode \(p\) and meeting \(D\) in nodes \(q_1\), \(q_2\). They also show that \(\overline{M}_g^{\mathrm{hs}}\) is not \(\mathbb Q\)-factorial for \(g \geq 7\) and propose, conjecturally, a way to eliminate this pathology.NEWLINENEWLINEFor the entire collection see [Zbl 1236.14001].