Abstract: Often in Algebraic Geometry linear systems will have an "expected dimension" which should hold most of the time, the problem then becomes to study when this expectation fails. In this talk we discuss a version of this problem. In particular we introduce the concept of a "very unexpected hypersurface" passing through a fixed set of points Z. These occur when Z imposes less than the "expected" number of conditions on the ideal sheaf of a generic linear subspace. We show in certain cases these can be characterized via combinatorial data and geometric data from the Hyperplane Arrangement dual to Z. We close by discussing relationships between this problem and certain motivating problems in Combinatorics, Matroid Theory and Hyperplane Arrangements.