Line arrangements and configurations of points with an unusual geometric property

The SHGH conjecture proposes a solution to the question of how many conditions a general union of fat points imposes on the complete linear system of curves in P² of fixed degree d, and it is known to be true in many cases. We propose a new problem, namely to understand the number of conditions imposed by a general union of fat points on the incomplete linear system defined by the condition of passing through a given finite set of points Z (not general). Motivated by work of Di Gennaro-Ilardi-Vallès and Faenzi-Vallès, we give a careful analysis for the case where there is a single general fat point, which has multiplicity d−1. There is an expected number of conditions imposed by this fat point, and we study those Z for which this expected value is not achieved. We show, for instance, that if Z is in linear general position then such unexpected curves do not exist. We give criteria for the occurrence of such unexpected curves and describe the range of values of d for which they occur. Unexpected curves have a very particular structure, which we describe, and they are often unique for a given set of points. In particular, we give a criterion for when they are irreducible, and we exhibit examples both where they are reducible and where they are irreducible. Furthermore, we relate properties of Z to properties of the arrangement of lines dual to the points of Z. In particular, we obtain a new interpretation of the splitting type of a line arrangement. Finally, we use our results to establish a Lefschetz-like criterion for Terao's conjecture on the freeness of line arrangements.