If \(P\) is a point of \(\mathbb{P}^2\), denote by \(mP\) the \((m-1)\)-st infinitesimal neighborhood of \(P\), defined by the saturated ideal \(I_P^m\). This scheme \(mP\) is called the fat point with multiplicity \(m\) supported at \(P\). This paper considers a zero-dimensional scheme of the type \(Z = \bigcup_{1 \leq i \leq r} m_i P_i\) and studies the possible Hilbert functions and minimal free resolutions of such schemes. This question is still open, even for a general choice of the \(P_i\), when the \(m_i\) are not all equal. For a good general discussion of this topic, with many references see B. Harbourne, “Problems and Progress: A survey on fat points in \(\mathbb{P}^2\)”, available at http://www.math.unl.edu/~bharbour. The case where \(m_i=2\) for all \(i\) (“double points”) is treated by M. Idà [J. Algebra 216, No. 2, 741-753 (1999; Zbl 0943.13008)].
The current paper studies the cohomology of the sheaves \(\Omega_{\mathbb{P}^2} \otimes {\mathcal I}_Z(t)\). The authors apply this to the study of general unions of double points, giving a new proof of the result of Idà as well as some new results. In particular, they show how to find configurations of double points which fail to have the expected cohomology in prescribed ways. As a result, the authors show that the configurations can have the “expected” Hilbert function without having the “expected” number of minimal generators, and they describe other Hilbert functions that occur. They also give conditions which force \(Z\) to be supported on a line. They end with some results about fat points of higher multiplicities.