A birational contraction of genus 2 tails in the moduli space of genus 4 curves I (Q2878741)

Brendan Hassett initiated the fascinating program of applying the minimal model program to the moduli space of stable curves by computing the sequence of log canonical models of \(\overline{\mathcal{M}}_g\) associated to a rational multiple \(\alpha\) of the total boundary divisor and studying the flips and divisorial contractions arising in this process. The spaces NEWLINE\[NEWLINE\overline{\mathcal{M}}_g(\alpha) := \mathrm{Proj} \bigoplus_{m \geq 0} H^0(\overline{\mathcal{M}}_g,m(K+\alpha D))NEWLINE\]NEWLINE start when \(\alpha=1\) by recovering the Deligne-Mumford compactification, since a classic result is that \(D\) is ample on \(\overline{\mathcal{M}}_g\), then one decreases \(\alpha\) as far as one can. For large enough genus (e.g., \(g = 22\) or \(g \geq 24\)) this moduli space is of general type so the sequence terminates with \(\alpha = 0\) being the canonical birational model of \(\overline{\mathcal{M}}_g\), a tantalizing object believed to be a moduli space of highly singular curves but whose basic properties and concrete description remain a mystery. This program is often called the Hassett-Keel program. For small genus the log canonical models end up being a point long before \(\alpha\) reaches zero, so rather than describing the (uninteresting) canonical model, the goal is simply to describe the entirely of the finitely many steps in the program until the last model is indeed a point. In fact, Hassett launched this program with his wonderful paper [\textit{B. Hassett}, Prog. Math. 235, 169--192 (2005; Zbl 1094.14017)] detailing the case \(g=2\). The big results then came in papers of \textit{B. Hassett} and \textit{D. Hyeon} [Trans. Am. Math. Soc. 361, No. 8, 4471--4489 (2009; Zbl 1172.14018), and Ann. Math. (2) 177, No. 3, 911--968 (2013; Zbl 1273.14034)] which described the first divisorial contraction and the first flip in all genera.NEWLINENEWLINESubsequently, work on the program bifurcated in some sense. Some authors wanted to push the complete description further in low genus bigger than 2, and others (notably a group including Alper, Fedorchuk, Smyth) pushed the general investigation further by employing machinery relying heavily on stacks and a careful understanding of the curve singularities that arise as one proceeds. For instance, in a recent preprint they described the second flip. As for low genus, the present article discusses the case genus 4, which was also treated contemporaneously by \textit{S. Casalaina-Martin} et al. [J. Algebr. Geom. 23, No. 4, 727--764 (2014; Zbl 1327.14207)] and which builds on earlier work of \textit{M. Fedorchuk} [Int. Math. Res. Not. 2012, No. 24, 5650--5672 (2012; Zbl 1258.14032)] where the final step of the program was described. The final step of the program for genus 6 was also worked out recently, by \textit{F. Müller} [Algebra Number Theory 8, No. 5, 1113--1126 (2014; Zbl 1323.14018)]. The low genus approach (and much of the general genus approach too) is based heavily on GIT techniques. A nice survey article discussing aspects of this was written by \textit{J. Alper} and \textit{D. Hyeon} [Contemp. Math. 564, 87--106 (2012; Zbl 1271.14033)].