The geometry of points on quantum projectivizations

Abstract

Suppose $S$ is an affine, noetherian scheme, $X$ is a separated, noetherian $S$-scheme, $\mathcal{E}$ is a coherent ${\mathcal{O}}_X$-bimodule and $\mathcal{I}\subset T(\mathcal{E})$ is a graded ideal. We study the geometry of the functor $\hat{I}{“}_n$ of flat families of truncated $\mathcal{B}=T(\mathcal{E})/\mathcal{I}$-point modules of length $n+1$. We then use the results of our study to show that the point modules over $\mathcal{B}$ are parameterized by the closed points of ${\mathbb{P}}_{X^2}(\mathcal{E})$. When $X={\mathbb{P}}^1$, we construct, for any $\mathcal{B}$-point module, a graded ${\mathcal{O}}_X-\mathcal{B}$-bimodule resolution.

25 pages