  We give a geometric invariant theory (GIT) construction of the log canonical
model $\bar M_g(\alpha)$ of the pairs $(\bar M_g, \alpha \delta)$ for $\alpha
\in (7/10 - \epsilon, 7/10]$ for small $\epsilon \in \mathbb Q_+$. We show that
$\bar M_g(7/10)$ is isomorphic to the GIT quotient of the Chow variety
bicanonical curves; $\bar M_g(7/10-\epsilon)$ is isomorphic to the GIT quotient
of the asymptotically-linearized Hilbert scheme of bicanonical curves. In each
case, we completely classify the (semi)stable curves and their orbit closures.
Chow semistable curves have ordinary cusps and tacnodes as singularities but do
not admit elliptic tails. Hilbert semistable curves satisfy further conditions,
e.g., they do not contain elliptic bridges. We show that there is a small
contraction $\Psi: \bar M_g(7/10+\epsilon) \to \bar M_g(7/10)$ that contracts
the locus of elliptic bridges. Moreover, by using the GIT interpretation of the
log canonical models, we construct a small contraction $\Psi^+ : \bar
M_g(7/10-\epsilon) \to \bar M_g(7/10)$ that is the Mori flip of $\Psi$.
