  In order to determine the Hilbert function of the ideal of a fat point
subscheme of projective space, we show that it is enough to determine, both for
the subscheme itself and the subschemes obtained from it by successively
adjoining to it additional general points, the least degrees in which the
ideals of the subschemes are nonzero. This point of view gives rise to the
following conjecture: if C is any reduced irreducible curve on a surface X
obtained by blowing up P^2 at any finite number of generic points, then C^2 >
g-2, where g is the arithmetic genus of C. This conjecture is shown to be
equivalent to the usual well-known conjectures for computing dimensions of
complete linear systems on X.
