Actions on the Picard group of smooth Fano threefolds

TL;DR

This work classifies the possible images of the automorphism action on the Picard group for smooth Fano threefolds by comparing $AutP(X)$ with Matsuki’s Weyl group $WG_X$ and exploiting blow-up, divisor, double-cover, and toric descriptions. The authors develop and apply a framework based on the lifting of automorphisms to the total space of $-K_X$, the KKMR movable cone decomposition, and explicit geometric constructions to realize or bound $AutP(X)$ for each deformation family. A central result is that $AutP(X)=WG_X$ for most families, with a short list of exceptions where the realizability is confirmed on suitably chosen smooth members or where the bound is known not to be attained (e.g., family № 2.2). They also show that $H^1(G, ext{Pic}(X))=0$ for all $ ho\le 5$, providing a cohomological obstructions-free landscape for these finite-group actions, and present a table summarizing the entire classification and related cohomological data. The results have implications for equivariant birational geometry and arithmetic obstructions tied to group actions on Picard lattices of Fano threefolds.

Abstract

We classify the possible images of the action of the group of automorphisms of a smooth Fano threefold on its Picard group. We also study the first group cohomology of the Picard group for families of smooth Fano threefolds.

Theorems & Definitions (65)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Proposition 2.1
• proof
• Lemma 2.2
• Remark 2.3
• Theorem 3.1
• Definition 3.2
• Proposition 3.3