Moduli spaces for point modules on naïve blowups

Abstract

The naïve blowup algebras developed by Keeler, Rogalski, and Stafford, after examples of Rogalski, are the first known class of connected graded algebras that are noetherian but not strongly noetherian. This failure of the strong noetherian property is intimately related to the failure of the point modules over such algebras to behave well in families: puzzlingly, there is no fine moduli scheme for such modules although point modules correspond bijectively with the points of a projective variety $X$. We give a geometric structure to this bijection and prove that the variety $X$ is a coarse moduli space for point modules. We also describe the natural moduli stack $X_{\infty }$ for embedded point modules — an analog of a “Hilbert scheme of one point” — as an infinite blowup of $X$ and establish good properties of $X_{\infty }$. The natural map $X_{\infty }\to X$ is thus a kind of “Hilbert–Chow morphism of one point" for the naïve blowup algebra.