The (unexpected) importance of knowing $\alpha$

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.