Inversion of adjunction for $F$-signature

Abstract

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

21 pages, v3: Stronger main theorem (Removed the assumption that the Q-Gorenstein index be prime to the characteristic). Minor improvements in exposition