Resumen de Classifying Hilbert functions of fat point subschemes in P^2
Anthony V. Geramita, Brian Harbourne, Juan C. Migliore
The paper~\cite{GMS} raised the question of what the possible Hilbert functions\break are for fat point subschemes of the form 2$p_1+\cdots+$ 2$p_r$, for all possible choices of~$r$ distinct points in~$\pr2$. We study this problem for $r$ points in~$\pr2$ 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~\cite{{freeres}, {BHProc}} 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,\ldots,p_r$ and $p'_1,\ldots,p'_r$ have the same {\tentimesit configuration type} if for all choices of nonnegative integers $ m_i$, $ Z=m_1p_1+\cdots+m_rp_r$ and $Z'=m_1p'_1+\cdots+m_rp'_r$ have the same Hilbert function.) Assuming either that~7 $\le r\le$~8 (see~\cite{GuH} 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 $Z=m_1p_1+\cdots+m_rp_r$. 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.
