Classifying Hilbert functions of fat point subschemes in $\mathbb P^2$

A recent paper by the first and third authors together with Sabourin raised the question of what the possible Hilbert functions are for fat point subschemes of the form $2p_1+...+2p_r$, for all possible choices of $r$ distinct points in the projective plane. We study this problem for $r$ points in the plane over an algebraically closed field $k$ of arbitrary characteristic in case either $r \le 8$ or the points lie on a (possibly reducible) conic. In either case, it follows from work of the second author that there are only finitely many configuration types of points, where our notion of configuration type is a generalization of the notion of a representable combinatorial geometry, also known as a representable simple matroid. (We say $p_1,...,p_r$ and $p'_1,...,p'_r$ have the same {\it configuration type} if for all choices of nonnegative integers $m_i$, $Z=m_1p_1+...+m_rp_r$ and $Z'=m_1p'_1+...+m_rp'_r$ have the same Hilbert function.) Assuming either that $7 \le r\le 8$ (see recent work of Guardo and the second author for the cases $r\le 6$) or that the points $p_i$ lie on a conic, we explicitly determine all the configuration types, and show how the configuration type and the coefficients $m_i$ determine (in an explicitly computable way) the Hilbert function (and sometimes the graded Betti numbers) of $Z=m_1p_1+...+m_rp_r$. We demonstrate our results by explicitly listing all Hilbert functions for schemes of $r\le 8$ double points, and for each Hilbert function we state precisely how the points must be arranged (in terms of the configuration type) to obtain that Hilbert function.

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