Let \(p_1,\dots,p_n\) be general points in the projective plane \(\mathbb P^2\), with corresponding homogeneous ideals \(P_1,\dots,P_n\). An ideal of the form \(I({\mathbf m},n) = P_1^{m_1} \cap \dots \cap P_n^{m_n}\) (where \(\mathbf m = (m_1,\dots,m_n)\) is an \(n\)-tuple of non-negative integers) defines a fat point subscheme \(Z \subset \mathbb P^2\). If \(m_1 = \dots = m_n = m\) (say), we say that m (or \(Z\)) is uniform and just write \(I(m,n)\). There are several conjectures about the Hilbert function and minimal free resolution of the ideal \(I(\mathbf m,n)\), and the answers are known in some cases (mostly uniform cases) and counterexamples are known in a few cases. However, all known failures involve \(n < 9\). The conjectures all involve some sort of maximal rank property.
In this paper the authors introduce the notion of quasiuniformity, saying that \(\mathbf m\) is quasiuniform if \({\mathbf m} = (m_1,m_2,\dots,m_n)\) with \(n \geq 9\) and \(m_1 = \dots = m_9 \geq m_{10} \geq \dots \geq m_n \geq 0\). They give conjectures for the Hilbert function and minimal free resolution of quasiuniform fat point schemes. Since quasiuniformity is an extension of the notion of uniformity, they note that their conjectures contain as special cases the existing conjectures (or in a few cases, theorems) for the uniform case. They also prove a number of solid results that give substantial evidence for their conjectures, especially for the uniform case. In particular, they prove the conjectures for infinitely many \(m\) for each of infinitely many \(n\), and for infinitely many \(n\) for every \(m>2\). They also show that in many cases the Hilbert function conjecture implies the resolution conjecture. As a by-product of their work, they get a strong bound on the regularity of \(I(m,n)\).