The geometry of points on quantum projectivizations

The use of geometric invariants has recently played an important role in the solution of classification problems in noncommutative ring theory. We construct geometric invariants of noncommutative projectivizations, a significant class of examples in noncommutative algebraic geometry. More precisely, if S is an affine, noetherian scheme, X is a separated, noetherian S-scheme, $E$ is a coherent $OX$-bimodule and $I\subset TE$ is a graded ideal then we develop a compatibility theory on adjoint squares in order to construct the functor Γ_n of flat families of truncated $TE/I$-point modules of length n + 1. For n ≥ 1 we represent Γ_n as a closed subscheme of $PX2E\otimes n$. The representing scheme is defined in terms of both $In$ and the bimodule Segre embedding, which we construct. When $ProjTE/I$ is a quantum ruled surface, we study the point modules over $TE/I$. More precisely, we use the geometry of the Γ_n 's to show that the point modules over $TE/I$ are parameterized by the closed points of $PX2E$. Furthermore, when $X=P1$, we construct, for any $TE/I$-point module, a graded $OX$ − $TE/I$-bimodule resolution.