The geometry of the ball quotient model of the moduli space of genus four curves (Q2895543)

S. Kondo defined a birational map between the moduli space of non-hyperelliptic curves of genus four and a ball quotient, thus providing a Baily-Borel compactification for the moduli space of such curves [\textit{S. Kondō}, Adv. Stud. Pure Math. 36, 383--400 (2002; Zbl 1043.14005)].NEWLINENEWLINEIn this paper the authors construct a GIT compactification \(\overline{M}_4^{\mathrm{GIT}}\) of the moduli space of non-hyperelliptic curves of genus four and study its relationship with Kondo's compactification. More precisely, the main theorem states that the natural period map between \(\overline{M}_4^{\mathrm{GIT}}\) and Kondo's compactification can be resolved by blowing-up one point (where the exceptional divisor parametrizes hyperelliptic curves with a \(g_2^1\)) and its resolution contracts a rational curve to one cusp.NEWLINENEWLINEThe space \(\overline{M}_4^{\mathrm{GIT}}\) is constructed by establishing a correspondence between genus four curves and cubic threefolds with an ordinary node and applying results by D. Allcock about GIT for cubic threefolds. Moreover, the authors prove that the space \(\overline{M}_4^{\mathrm{GIT}}\) coincides with a GIT quotient of the Chow variety of canonically embedded genus four curves and identify it with the Hassett-Keel space \(\overline{M}_4(5/9)\).NEWLINENEWLINESimilar results for the moduli space of curves of genus three have been given by \textit{E. Looijenga} [Contemp. Math. 422, 107--120 (2007; Zbl 1126.14034)] and the reviewer [Nagoya Math. J. 196, 1--26 (2009; Zbl 1184.14060)].NEWLINENEWLINEFor the entire collection see [Zbl 1236.14001].