$k\geq 9$. Since $J(8/7)=9$, the result follows.

As explained in Section \[sec:strategy\], we can combine the above results with Remark \[rmk:AutomorphismsTrivialInitialStep\] to find obstructing exceptional classes in $H_{2}(X_{2p+2};{\mathbb{Z}})$. These results can be easily translated into curves in the $2p+1$-fold blow-up of $M_{\mu}^{0}$ as well by using the identification of the two spaces.

\[cor:TrivialOddCaseObstructionClasses\] The exceptional classes in $H_{2}(X_{2p+2};{\mathbb{Z}})$ that give the obstructions to the embedding of $2p+1$ balls into $M_{\mu}^{0}$ are of type $$\begin{aligned}
  \left( 1\,; 1^{\times 2}, 0^{\times (2p-1)}\right) 
    & \text{~when~} \mu\in\left[ p+1,~ \infty \right);\\
  \left( p\,; p-1, 1^{\times 2p}, 0\right) 
    & \text{~when~} \mu\in\left[ p+1-\sqrt{2p+1},~ p+1 \right).\end{aligned}$$ For $\mu\in\left[ 1, p+1-\sqrt{2p+1} \right)$, the only obstruction is given by the volume condition.

On each interval, the reduction algorithm defines an automorphism $\phi\in D_{K}(1,k+1)$ as a composition of Cremona moves and reorderings. The obstructing exceptional class is then given by $\phi^{*} E_{k+1}$. In the present cases, the automorphism is $$\phi =
\begin{cases}
(\mathfrak{RC})^{p} 
    & \text{~for~} \mu \in \left[ p+1,~ \infty \right)\\
(\mathfrak{R}\mathfrak{C})(\mathfrak{S}\mathfrak{C})(\mathfrak{R}\mathfrak{C})^{p-2}
    & \text{~for~} \mu \in \left[ p+1-\sqrt{2p+1},~ p+1 \right)
\end{cases}$$ where $\mathfrak{R}$ and $\mathfrak{C}$ are the automorphisms defined in Remark \[rmk:AutomorphismsTrivialInitialStep\], and where $\mathfrak{S}$ corresponds to the permutation $(1,4,2,3,5,\ldots,k+2)$.

The even case $k=2p$
--------------------

This subsection is dedicated to proving 