Naive blowups and canonical birationally commutative factors

TL;DR

This paper extends the understanding of canonical birationally commutative factors of noetherian graded algebras, showing under certain conditions they relate to naive blowups and have noetherian point spaces.

Contribution

It generalizes previous results to a broader class of noetherian algebras, establishing a connection to naive blowups and properties of point modules.

Findings

Existence of a homomorphism from R to a naive blowup algebra.

The image of R under this homomorphism is finite-dimensional and satisfies a universal property.

The point space Y is proven to be noetherian.

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.