Correction to “Combinatorics and geometry of power ideals”: Two counterexamples for power ideals of hyperplane arrangements

Abstract:

We disprove Holtz and Ron’s conjecture that the power ideal $C_{\mathcal {A},-2}$ of a hyperplane arrangement $\mathcal {A}$ (also called the internal zonotopal space) is generated by $\mathcal {A}$-monomials. We also show that, in contrast with the case $k \geq -2$, the Hilbert series of $C_{\mathcal {A},k}$ is not determined by the matroid of $\mathcal {A}$ for $k \leq -6$.