Vol. 7, No. 4, 2013

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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

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 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. The natural map X X is thus a kind of “Hilbert–Chow morphism of one point" for the naïve blowup algebra.

naïve blowup, point module, point space
Mathematical Subject Classification 2010
Primary: 16S38
Secondary: 16D70, 16W50, 14A20, 14D22
Received: 28 October 2010
Revised: 6 April 2012
Accepted: 5 November 2012
Published: 29 August 2013
Thomas A. Nevins
Department of Mathematics
University of Illinois at Urbana–Champaign
1409 West Green Street
Urbana, IL
United States
Susan J. Sierra
School of Mathematics
The University of Edinburgh
James Clerk Maxwell Building
The King’s Buildings
Mayfield Road
United Kingdom