#### Vol. 7, No. 4, 2013

 Download this article For screen For printing
 Recent Issues
 The Journal Cover Editorial Board Editors' Addresses Editors' Interests About the Journal Scientific Advantages Submission Guidelines Submission Form Subscriptions Editorial Login Contacts Author Index To Appear ISSN: 1944-7833 (e-only) ISSN: 1937-0652 (print)
Moduli spaces for point modules on naïve blowups

### Thomas A. Nevins and Susan J. Sierra

Vol. 7 (2013), No. 4, 795–834
##### 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.

##### Keywords
naïve blowup, point module, point space
##### Mathematical Subject Classification 2010
Primary: 16S38
Secondary: 16D70, 16W50, 14A20, 14D22
##### Milestones
Received: 28 October 2010
Revised: 6 April 2012
Accepted: 5 November 2012
Published: 29 August 2013
##### Authors
 Thomas A. Nevins Department of Mathematics University of Illinois at Urbana–Champaign 1409 West Green Street MC-382 Urbana, IL 61801 United States Susan J. Sierra School of Mathematics The University of Edinburgh James Clerk Maxwell Building The King’s Buildings Mayfield Road Edinburgh EH9 3JZ United Kingdom http://www.maths.ed.ac.uk/~ssierra/