  We consider the open problem of determining the graded Betti numbers for fat
point subschemes supported at general points of the projective plane. We relate
this problem to the open geometric problem of determining the splitting type of
the pullback of the cotangent bundle on the plane to the normalization of
certain rational plane curves. We give a conjecture for the graded Betti
numbers which would determine them in all degrees but one for every fat point
subscheme supported at general points of the plane. We also prove our Betti
number conjecture in a broad range of cases. An appendix discusses many more
cases in which our conjecture has been verified computationally and provides a
new and more efficient computational approach for computing graded Betti
numbers in certain degrees. It also demonstrates how to derive explicit
conjectural values for the Betti numbers and how to compute splitting types.
