Naive blowups and canonical birationally commutative factors

Abstract

In 2008, Rogalski and Zhang showed that if R is a strongly noetherian connected graded algebra over an algebraically closed field, then R has a canonical birationally commutative factor. This factor is, up to finite dimension, a twisted homogeneous coordinate ring B(X, L, s); here X is the projective parameter scheme for point modules over R, as well as tails of points in qgr-R. (As usual, s is an automorphism of X, and L is an s-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 k-algebra, where k is an uncountable algebraically closed field. Let Y denote the parameter space (or stack or proscheme) parameterizing R-point modules, and suppose there is a projective variety X that is a coarse moduli space for tails of points. There is a canonical map p: Y -> 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 -> B(X, L, s), and that g(R) is, up to finite dimension, a naive blowup on X in the sense of Keeler, Rogalski, and Stafford, and satisfies a universal property. We further show that the point space Y is noetherian.