F-SIGNATURE UNDER BIRATIONAL MORPHISMS

We study F-signature under proper birational morphisms pi : Y -> X, showing that F-signature strictly increases for small morphisms or if K-Y <= pi*K-X. In certain cases, we can even show that the F-signature of Y is at least twice as that of X. We also provide examples of F-signature dropping and Hilbert Kunz multiplicity increasing under birational maps without these hypotheses.