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- W2591719137 abstract "In 2008, Rogalski and Zhang (Math Z 259(2):433–455, 2008) showed that if R is a strongly noetherian connected graded algebra over an algebraically closed field $${Bbbk }$$ , then R has a canonical birationally commutative factor. This factor is, up to finite dimension, a twisted homogeneous coordinate ring $$B(X, mathcal {L}, sigma )$$ ; here X is the projective parameter scheme for point modules over R, as well as tails of points in $${{mathrm{Qgr-!}}}R$$ . (As usual, $$sigma $$ is an automorphism of X, and $$mathcal {L}$$ is a $$sigma $$ -ample invertible sheaf on X). We extend this result to a large class of noetherian (but not strongly noetherian) algebras. Specifically, let R be a noetherian connected graded $${Bbbk }$$ -algebra, where $${Bbbk }$$ is an uncountable algebraically closed field. Let $$Y_infty $$ denote the parameter space (or stack or proscheme) parameterizing R-point modules, and suppose there is a projective variety X that corepresents tails of points. There is a canonical map p: $$Y_infty rightarrow X$$ . If the indeterminacy locus of $$p^{-1}$$ is 0-dimensional and X satisfies a mild technical assumption, we show that there is a homomorphism g: $$R rightarrow B(X,mathcal {L},sigma )$$ , and that g(R) is, up to finite dimension, a naive blowup on X in the sense of Keeler et al. (Duke Math J 126(3):491–546, 2005), Rogalski and Stafford (J Algebra 318(2):794–833, 2007) and satisfies a universal property. We further show that the point space $$Y_infty $$ is noetherian." @default.
- W2591719137 created "2017-03-16" @default.
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- W2591719137 date "2015-05-26" @default.
- W2591719137 modified "2023-09-27" @default.
- W2591719137 title "Naïve blowups and canonical birationally commutative factors" @default.
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- W2591719137 doi "https://doi.org/10.1007/s00209-015-1470-3" @default.
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