for $1 \leq k \leq \ell-1$.

Note that the dimension of the star locus is $$\begin{aligned}
\dim(S) = \text{Max}_{1 \leq i \leq \ell-1} d - \left( (m_{\ell-1}+1)m_\ell + m_{i-1}m_i + \sum_{j=i}^{\ell-1} m_j \right)\end{aligned}$$

For each choice of $m$ integers $1 \leq k_1 \leq \cdots \leq k_m \leq \ell-1$, we define the locus $C_{k_1, \ldots, k_m}$ by the following formula: $$\begin{gathered}
C_{k_1,\ldots,k_m}
=
\left\{\left.
p = (M_1,b_1,\ldots,M_\ell,b_\ell)
\:\right|\:
\eqref{eq:C_conds}
\right\},
\nonumber
\\
M_\ell = 0,
b_\ell = \sum_{\alpha=1}^n y_\alpha,
M_{k_1}=\cdots=M_{k_m}=0,
b_{k_1}=\cdots=b_{\ell-1}=0.
\label{eq:C_conds}\end{gathered}$$

For each $m$, we define the $m$-core $C_m$ as the union over all $m-$tuples $k_1, ..., k_m$ of $C_{m_1, ..., m_k}$. Note that the star locus $S$ is equal to the $1$-core $C_1$.

Note that

$$(0,...,0) = C_\ell \subset ... C_k \subset C_{k-1} ... C_2 \subset C_1 = S$$ and that each $C_k$ is the union of linear subspaces. We call $C_2$ the core locus, and denote it by $C$. This locus will be of particular interest to us, and it contains all the higher cores $C_3, C_4$, and so on.

\[starcore\]

![The star locus is a union of linear spaces, and looks something like the blue set in this image. The core locus is a subset of the 