Dual F-signature

I will report on the joint work with Kevin Tucker which completed the basics of the theory of dual F-signature/F-rational signature. 

F-signature is a volume-like invariant of singularities of positive characteristic: its minimal value, 0, characterises the class of strongly F-regular singularities and its maximal value, 1, detects non-singularity. Hence, F-signature stratifies the class of strongly F-regular singularities. F-signature is now reasonably well-understood and has strong applications in algebraic geometry, for example, giving an explicit bound on the size of the local étale fundamental group.

The class of F-rational singularities is more severe than the class of F-regular singularities, but is closely related. For example, after work of Smith, Vélez, and Blickle one can think about F-rationality as F-regularity of the dualising module. Due to the close connection, it was expected that there should an F-signature like invariant for F-rationality and two distinct definitions were given: dual F-signature by Sannai and F-rational signature of Hochster — Yao. 

The main result of our work is that, after a normalization, the Hochster—Yao signature is equal to Sannai’s. This allows combination of the techniques that results in a theory almost parallel to that of F-signature.