subspace arrangement not generated by products of linear forms

Question

Let $k$ be an algebraically closed field. A subspace arrangement in $k^d$ is a finite union of proper subspaces of $k^d$ and similarly we define an arrangement of hyperplanes. Let $A$ be a subspace arrangement and $I_A$ its vanishing ideal. Note that $I_A$ is homogeneous. H. Derksen http://arxiv.org/abs/math/0510584 proved that if there are $n$ subspaces in $A$, then all homogeneous components $I_{A,k}$ of $I_A$ are generated by products of linear forms that vanish on $A$ for $k\ge n$. Note that each product of linear forms vanishing on $A$ defines a hyperplane arrangement that contains $A$. I am interested in finding a counterexample for the case $k<n$, i.e. is there a subspace arrangement that is not generated by products of linear forms? In this paper http://arxiv.org/abs/math/0401373 an algorithm is given that determines whether $I_A$ is generated by products of linear forms, however i can not detect any counterexample.