  By defining a fat point subscheme of $P^2$ to be a 0-dimensional subscheme
defined by a sheaf of integrally closed ideals one extends the notion of fat
point subschemes to allow infinitely near points. With this notion of fat
points, this preprint determines the number of generators in each degree in a
minimal homogeneous set of generators for the homogeneous ideal defining any
fat point subscheme supported at points of any plane conic, smooth or not.
Special cases for points on a cubic are also studied. All work is over an
algebraically closed field of arbitrary characteristic.
