Resolutions of small sets of fat points

We investigate the minimal graded free resolutions of ideals of at most n+1 fat points in general position in image. 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 image, verifying a conjecture of Fatabbi, and at most n+1 general double points in image.