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The naive blow-up algebras developed by Keeler-Rogalski-Stafford, afterexamples of Rogalski, are the first known class of connected graded algebrasthat are noetherian but not strongly noetherian. This failure of the strongnoetherian property is intimately related to the failure of the point modulesover such algebras to behave well in families: puzzlingly, there is no finemoduli scheme for such modules, although point modules correspond bijectivelywith the points of a projective variety X. We give a geometric structure tothis bijection and prove that the variety X is a coarse moduli space for pointmodules. We also describe the natural moduli stack \tilde{X} for embedded pointmodules---an analog of a "Hilbert scheme of one point"---as an infinite blow-upof X and establish good properties of \tilde{X}. The natural map \tilde{X} -> Xis thus a kind of "Hilbert-Chow morphism of one point" for the naive blow-upalgebra.

Moduli spaces for point modules on naive blowups