Mats Boij: Cones of Hilbert functions

I will report on a joint work with Gregory G. Smith where we studied the closed convex hull of various collections of Hilbert functions. In this work we focused on graded modules generated in degree zero over a standard graded polynomial ring. In this context, we completely described the supporting hyperplanes and extreme rays for the cones generated by the Hilbert functions of all modules, all modules with bounded \(a\)-invariant, and all modules with bounded Castelnuovo-Mumford regularity. The first of these cones is infinite-dimensional and simplicial, the second is finite-dimensional but neither simplicial nor polyhedral, and the third is finite-dimensional and simplicial. We also gave explicit linear bounds for the graded Betti numbers of modules with a given Hilbert function in the case of bounded Castelnuovo-Mumford regularity.