Moduli spaces for point modules on naïve blowups

Nevins, Thomas; Sierra, Susan

doi:10.2140/ant.2013.7.795

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Moduli spaces for point modules on naïve blowups

Thomas A. Nevins and Susan J. Sierra

Vol. 7 (2013), No. 4, 795–834

DOI: 10.2140/ant.2013.7.795

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/

Vol. 7, No. 4, 2013