The Points of Quadratic Algebras
Abstract: I plan to present recent joint work by Brad Shelton and myself, with emphasis on the following counter-intuitive result. Let A denote a noncommutative algebra on four generators with six defining relations (each homogeneous of degree 2), and let $\Gamma$ denote the locus of zeros of the defining relations of A. If $\Gamma$ is finite, then the space of (1,1)-forms that vanish on $\Gamma$ is the span of the defining relations of A. The result concerns the ``points'' of A and has a counterpart involving ``lines'' of A. While the results are noncommutative in nature, the proofs use only commutative algebra.