Moduli spaces for point modules on naive blowups

TL;DR

This paper studies the geometric structure of point modules over naive blow-up algebras, showing that the variety X acts as a coarse moduli space and describing a related moduli stack as an infinite blow-up.

Contribution

It establishes that the variety X is a coarse moduli space for point modules and constructs a moduli stack as an infinite blow-up, linking algebraic properties to geometric structures.

Findings

X is a coarse moduli space for point modules

The moduli stack itilde{X} is an infinite blow-up of X

The map itilde{X}  X acts as a 'Hilbert-Chow' morphism

Abstract

The naive blow-up algebras developed by Keeler-Rogalski-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 \tilde{X} for embedded point modules---an analog of a "Hilbert scheme of one point"---as an infinite blow-up of X and establish good properties of \tilde{X}. The natural map \tilde{X} -> X is thus a kind of "Hilbert-Chow morphism of one point" for the naive blow-up algebra.