# The theory of F-rational singularities

# The theory of F-rational singularities

**In-Person Talk **

There are a number of invariants defined via Frobenius in the study of singularities in characteristic. One such is the F-signature, which can be viewed as a quantitative measure of F-regularity – an important class of singularities central to the celebrated theory of tight closure pioneered by Hochster and Huneke, and closely related to KLT singularities via standard reduction techniques from characteristic zero. Recently, similar invariants have been introduced as a quantitative measures of F-rationality – another important class of F-singularity closely related to rational singularities in characteristic zero. These include the F-rational signature (Hochster-Yao), relative F-rational signature (Smirnov-Tucker), and dual F-signature (Sannai). In this talk, I will discuss new results in joint work with Smirnov relating each of these invariants. In particular, we show that the relative F-rational signature and dual F-signature coincide, while also verifying that the dual F-signature limit converges.