Vector bundles and tight closure (Triest 2023)

These are lecture notes for the course by Holger Brenner on vector bundles and tight closure held in Trieste in May 2023 within the school on Commutative Algebra and Algebraic Geometry in Prime Characteristics. For the school see [1].

We introduce vector bundles on smooth projective curves, in particular the notion of semistability and its variant in positive characteristic, strong semistability. We explain how these notions provide tools to study questions from Hilbert-Kunz-theory and tight closure in the corresponding two-dimensional homogeneous coordinate rings. In particular we will discuss rationality of Hilbert-Kunz multiplicity, relation between tight closure and plus closure, behaviour under change of prime numbers and the localization problem. Time permitting, we will also mention typical new problems arising in higher dimension.

Lecture 1 - Hilbert-Kunz theory and vector bundles

Lecture 2 - Tight closure and torsors

Lecture 3 - Plus closure and trivializable bundles

Lecture 4 - Deformations and localization problem

For the recorded lectures see Lecture 1, Lecture 2, Lecture 3, Lecture 4. For the pdf-version of the lectures see file:TriestBrennerlecturenotes.pdf.