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We show that for $\alpha \in (2/3, 7/10)$, the log canonical model $\barM_4(\alpha)$ of the pair $(\bar M_4, \alpha \delta)$ is isomorphic to themoduli space $\bar M_4^{hs}$ of h-semistable curves, and that there is abirational morphism $\Xi: \bar M_4^{hs} \to \bar M_4(2/3)$ that contracts thelocus of curves $C_1\cup_p C_2$ consisting of genus two curves meeting in anode $p$ such that $p$ is a Weierstrass point of $C_1$ or $C_2$. To obtain thismorphism, we construct a compact moduli space $\bar M_{2,1}^{hs}$ of pointedgenus two curves that have nodes, ordinary cusps and tacnodes as singularity,and prove that it is isomorphic to Rulla's flip constructed in his thesis.

Birational contraction of genus two tails in the moduli space of genus four curves I