The ICTP school and workshop will introduce commutative algebra and algebraic geometry in prime characteristics and their relationship with complex algebraic geometry. There have been many important developments in these fields during the last three decades. There will be four courses of lectures to highlight some of these developments related to vector bundles and tight closure, singularities in prime characteristics, F-signature and Hilbert-Kunz multiplicity and study of singularities via Big Cohen-Macaulay (BCM)-algebras, perfectoid spaces and Frobenius splitting.

School on Commutative algebra and algebraic geometry in prime characteristic (2-5 May, 2023)

Speaker: Holger Brenner, University of Osnabrueck, Germany

Title: Vector bundles and tight closure

Abstract: We introduce vector bundles on smooth projective curves, in particular the notion of semi-stability and its variant in positive characteristic, strong semi-stability. 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.

Speaker: Ilya Smirnov, Basque Centre for Applied Mathematics, Bilbao, Spain

Title: Singularity invariants in positive characteristic.

Abstract: F-signature and Hilbert-Kunz multiplicity are two of the most studied numerical invariants of singularities in positive characteristic. The goal of this class is to establish various algebraic and geometric properties of them and highlight the role of uniform convergence methods in the theory.

Speaker: Kevin Tucker, University of Illinois at Chicago, IL, USA

Title: Singularities via BCM Algebras

Abstract: In this lecture series, I will give an introduction to the use of Big Cohen Macaulay (BCM) algebras to study singularities in positive and mixed characteristic commutative algebra and algebraic geometry. Topics will include BCM algebras, splinter rings and the Direct Summand Theorem, and connections to tight closure and test ideals. We also aim to discuss recently introduced mixed characteristic perfectoid analogues of Hilbert-Kunz multiplicity and F-signature.

Speaker: Keiichi Watanabe, Nihon University, Tokyo, Japan

Title: F-singularities and their applications of characteristic p methods to singularity theory

Abstract: We introduce the notions of F-regular, F-rational, F-pure rings defined by characteristic p methods using tight closure and Frobenius splitting, and explain how these concepts are related to log terminal, rational, log canonical singularities which play important roles in MMP (Minimal Model Program) in algebraic geometry over fields of characteristic 0 using “reduction mod p” technique.

Organizing Committee

• Lothar Goettsch (ICTP, Trieste)

• Maria Evelina Rossi (University of Genoa, Genoa)

• Karen Smith (University of Michigan, Ann Arbor)

• Irena Swanson (Purdue University, West Lafayette )

• Bernd Ulrich (Purdue University, West Lafayette)

• Jugal Verma (IIT Bombay, Mumbai)

• Kei-ichi Watanabe (Nihon University, Tokyo)