  A rank one local system on the complement of a hyperplane arrangement is said
to be admissible if it satisfies certain non-positivity condition at every
resonant edges. It is known that the cohomology of admissible local system can
be computed combinatorially. In this paper, we study the structure of the set
of all non-admissible local systems in the character torus. We prove that the
set of non-admissible local systems forms a union of subtori. The relations
with characteristic varieties are also discussed.
