Quasi-Splines and their moduli
This work provides a new theoretical framework for splines in algebraic geometry, but the results are foundational and lack concrete applications or performance numbers, making it incremental for the field.
The authors introduce quasi-spline sheaves and related objects to study splines from a moduli theory perspective, proving that under certain hypotheses these objects admit fine moduli schemes, with the moduli of quasi-spline sheaves being proper and a natural compactification existing for ideal difference-conditions.
We study what we call quasi-spline sheaves over locally Noetherian schemes. This is done with the intention of considering splines from the point of view of moduli theory. In other words, we study the way in which certain objects that arise in the theory of splines can be made to depend on parameters. In addition to quasi-spline sheaves, we treat ideal difference-conditions, and individual quasi- splines. Under certain hypotheses each of these types of objects admits a fine moduli scheme. The moduli of quasi-spline sheaves is proper, and there is a natural compactification of the moduli of ideal difference-conditions. We include some speculation on the uses of these moduli in the theory of splines and topology, and an appendix with a treatment of the Billera-Rose homogenization in scheme theoretic language.