After recalling key notions defined last time, we introduce a complex whose homology is closely related to properties of the spline modules (Schenck-Stillman, based on a closely related complex of Billera). We describe what is known about these homology modules, and show some open questions (in both planar and higher dimension cases). After that, time permitting, we consider the properties of ideals generated by powers of linear forms, which are needed to understand dimensions of spline modules in higher degree/dimension. In the 2 variable case, a complete answer is known. In higher dimensions, the problem is wilder, but interesting. Using inverse systems of Macaulay (the person!), we relate this to so-called ideals of “fat points”.

(continued from February 29)