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 $s(R,\Delta )\ge s({\mathcal{O}}_D,{\mathrm{Diff}}_D(\Delta ))$. We also discuss the implications of our result for the conjectured connection between the $F$-signature and the normalized volume.

arXiv comment21 pages, v3: Stronger main theorem (Removed the assumption that the Q-Gorenstein index be prime to the characteristic). Minor improvements in exposition

References in corpus (11)

• F-signature exists
• F-singularities of pairs and Inversion of Adjunction of arbitrary codimension
• On the behavior of test ideals under finite morphisms
• A survey of test ideals
• Fundamental groups of $F$-regular singularities via $F$-signature
• On Strongly $F$-Regular Inversion of Adjunction
• Finite torsors over strongly $F$-regular singularities
• F-Signature of Affine Toric Varieties
• On the behavior of singularities at the $F$-pure threshold
• On the behavior of $F$-signatures, splitting primes, and test modules under finite covers
• $F$-purity deforms in $\mathbb{Q}$-Gorenstein rings

Cited by in corpus (1)

• Tame fundamental groups of pure pairs and Abhyankar's lemma