  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 $\mathbb P^2$ 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-Vall\`es and
Faenzi-Vall\`es, 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.
