  We investigate the minimal graded free resolutions of ideals of at most n+1
fat points in general position in P^n. Our main theorem is that these ideals
are componentwise linear. This result yields a number of corollaries, including
the multiplicity conjecture of Herzog, Huneke, and Srinivasan in this case. On
the computational side, using an iterated mapping cone process, we compute
formulas for the graded Betti numbers of ideals associated to two fat points in
P^n, verifying a conjecture of Fatabbi, and at most n+1 general double points
in P^n.
