Inversion of adjunction for $F$-signature

Abstract

Let $(R,\Delta+D)$ be a log $\mathbb{Q}$-Gorenstein pair where $R$ is a Noetherian, $F$-finite, normal, local domain of characteristic $p > 0$, $\Delta$ is an effective $\mathbb{Q}$-divisor and $D$ is an integral $\mathbb{Q}$-Cartier divisor. We show that the left derivative of the $F$-signature function $s(R,\Delta + tD)$ at $t = 1$ is equal to $-s(\mathcal{O}_D, \mathrm{Diff}_D(\Delta))$. This equality is interpreted as a quantitative form of inversion of adjunction for strong $F$-regularity. As an immediate corollary, we obtain the inequality $s(R,\Delta) \geq s(\mathcal{O}_D, \mathrm{Diff}_D(\Delta))$. We also discuss the implications of our result for the conjectured connection between the $F$-signature and the normalized volume.