Forcing Algebras, Syzygy Bundles, and Tight Closure

We give a survey about some recent work on tight closure and Hilbert-Kunz theory from the viewpoint of vector bundles. This work is based in understanding tight closure in terms of forcing algebras and the cohomological dimension of torsors of syzygy bundles. These geometric methods allowed to answer some fundamental questions of tight closure, in particular the equality between tight closure and plus closure in graded dimension two over a finite field and the rationality of the Hilbert-Kunz multiplicity in graded dimension two. Moreover, this approach showed that tight closure may behave weirdly under arithmetic and geometric deformations, and provided a negative answer to the localization problem.