Hilbert functions of fat point subschemes of the plane: the two-fold way
  Two approaches for determining Hilbert functions of fat point subschemes of $\mathbb P^2$ are demonstrated. A complete determination of the Hilbert functions which occur for 9 double points is given using the first approach, extending results obtained in a previous paper using the second approach. In addition the second approach is used to obtain a complete determination of the Hilbert functions for $n\geq 9$ $m$-multiple points for every $m$ if the points are smooth points of an irreducible plane cubic curve. Additional results are obtained using the first approach for $n\geq 9$ double points when the points lie on an irreducible cubic (but now are not assumed to be smooth points of the cubic). 
A.V. Geramita, B. Harbourne and J. Migliore
math.AG math.AC