  In this paper we present a smallest possible counterexample to the Numerical
Terao's Conjecture in the class of line arrangements in the complex projective
plane. Our example consists of a pair of two arrangements with $13$ lines.
Moreover, we use the newly discovered singular matroid realization spaces to
construct new examples of pairs of line arrangements having the same underlying
matroid but different free resolutions of the Milnor algebras. Such rare
arrangements are called Ziegler pairs in the literature.
