The final log canonical model of \(\overline{\mathcal{M}}_6\) (Q457325)

The highly influential Hassett-Keel program is to compute the canonical model of the moduli space of stable curves \(\overline{\mathcal{M}}_g\) by using the log minimal model program, or more specifically, by studying the spaces NEWLINE\[NEWLINE\overline{\mathcal{M}}_g(\alpha) := \mathrm{Proj}\bigoplus_{m \geq 0} H^0(m(D+\alpha K)),NEWLINE\]NEWLINE where \(K\) is the canonical divisor and \(D\) is the total boundary divisor on \(\overline{\mathcal{M}}_g\). It is a classical result that for \(\alpha=1\) we get the Deligne-Mumford compactification we started with (i.e., \(D+K\) is ample), so the program here is to cut a straight line segment path through the effective cone of \(\overline{\mathcal{M}}_g\) by starting with \(\alpha=1\) and hence the moduli space itself and concluding with \(\alpha=0\), which by definition is the canonical model. Despite the name, this program was initiated by \textit{B. Hassett} and \textit{D. Hyeon} [Ann. Math. (2) 177, No. 3, 911--968 (2013; Zbl 1273.14034)]. The beauty of the program so far has not been the end result, the canonical model itself (which thus far remains out of reach), but the journey itself: so far every flip and divisorial contraction arising in this \(\alpha\) path has admitted a modular interpretation, in which we change some aspect of the curves being parameterized. This suggests a very general and fascinating phenomenon linking moduli spaces with the minimal model program which has subsequently been explored in a variety of settings -- but it is here, with the moduli space of curves, that it all began.NEWLINENEWLINEAfter Hassett-Hyeon, the main program for arbitrary genus has come in a series of papers by Hyeon, Alper, Fedorchuk, Smyth, and van der Wyck, where an interesting blend of GIT, stack theory, positivity, and intersection theory are central to the story. Meanwhile, several authors (Jensen, Fedorchuck, et al.) have devoted effort to completing this program in low genus examples; here one does not seek the canonical model, since general type only holds for \(g \geq 22\), so instead one simply proceeds along the same \(\alpha\) path until the boundary of the effective cone is reached. The last big divisor obtained in this way corresponds to what is called the ``final'' log canonical model. The present paper studies this in the case \(g=6\). Having provided this background to the author's study, I leave it to the introduction of the paper itself to summarize the explicit description of this final model and some consequences he derives.