The F-Signature Function on the Ample Cone

Abstract For any fixed globally $F$-regular projective variety $X$ over an algebraically closed field of positive characteristic, we study the $F$-signature of section rings of $X$ with respect to the ample Cartier divisors on $X$. In particular, we define an $F$-signature function on the ample cone of $X$ and show that it is locally Lipschitz continuous. We further prove that the $F$-signature function extends to the boundary of the ample cone. We also establish an effective comparison between the $F$-signature function and the volume function on the ample cone. As a consequence, we show that for divisors that are nef but not big, the extension of the $F$-signature is zero.