Homogeneous fat point schemes in the projective plane

In regards to the Hilbert functions of fat point schemes in $Pn$, there are many open questions. In the case of fat point schemes in $P2$, where far more is known, there is not yet a classification theorem even for homogeneous double point schemes.; The spectrum of Hilbert functions possible for homogeneous fat point schemes in $P2$ corresponds to configurations of the points which range from the most singular, for example all the points lie on a line, to the most general setting, where the points are in general position. The latter situation is what we focus on in this paper. Although much progress has been made (see the Introduction) on understanding homogeneous fat points schemes in $P2$ with the points in general position, ‘most’ of the cases are still unresolved, at least for high multiplicity fat points.; For $P2$, there is a conjecture, called the Maximal Rank Conjecture, that predicts the Hilbert function of a scheme $X$ of N fat points of multiplicity m, provided N ≥ 10. The Maximal Rank Conjecture predicts that, if d = deg($X$), the Hilbert function of $X$ is the same as the Hilbert function of d simple points in general position, which is easy to calculate.; Recently, Alexander and Hirschowitz in [5] showed that for a fixed m, there is an N which depends on m, such that the conjecture holds for greater than or equal to N points of multiplicity m. Their proof gives no way, however, of computing how large N must be. The purpose of this paper is to provide a perspective from which, when the multiplicity is fixed, it becomes possible to explicitly compute values for N for which the Maximal Rank Conjecture holds.; One fact that makes the problem of computing the Hilbert function of fat points in general position interesting is that, if there exists a special configuration of N fat points of multiplicity m which has the Hilbert function predicted by the Maximal Rank Conjecture, then N points with multiplicity m in general position necessarily have the right Hilbert function.; Consequently, a way of attacking the problem of general points is to look for examples that have Hilbert functions that can easily be computed. The first 2 chapters, which analyze fat point schemes supported on lines, show that such configurations are good for producing examples of fat point schemes with a large variety of Hilbert functions.; My hope is that the results of chapters 1 and 2 will be successfully used and modified to give people an understanding of the Hilbert functions of fat point schemes beyond what is covered in this paper, both for general configurations and special ones.