  We consider the vanishing ideal of an arrangement of linear subspaces in a
vector space and investigate when this ideal can be generated by products of
linear forms. We introduce a combinatorial construction (blocker duality) which
yields such generators in cases with a lot of combinatorial structure, and we
present the examples that motivated our work. We give a construction which
produces all elements of this type in the vanishing ideal of the arrangement.
This leads to an algorithm for deciding if the ideal is generated by products
of linear forms. We also consider generic arrangements of points in ${\bf P}^2$
and lines in ${\bf P}^3.$
