Fano fourfolds with large anticanonical base locus

TL;DR

Let $X$ be a smooth Fano fourfold with $h^0(X,\mathcal{O}_X(-K_X))\ge 3$ and let the base locus $Bs(|-K_X|)$ contain an irreducible normal surface $B$. For a general anticanonical divisor $Y\in|-K_X|$, the authors decompose the restricted linear system as $-K_X|_Y\simeq M+B$ with $M$ mobile and show that, unless $M_X$ is nef (leading to a contradiction via a fibration argument), the base locus forces $B_X$ to behave so that an extension argument constrains sections and leads to $Y$ being non-$\mathbb{Q}$-factorial. The core strategy splits into a nef vs not-nef analysis of $M_X$, employing positivity, Kawamata–Viehweg vanishing, and inversion-of-adjunction to control singularities, together with a Mayer-type reasoning for Calabi–Yau-type settings. The resulting theorem demonstrates that large anticanonical base loci in fourfolds induce moving singularities in the anticanonical system and provides a framework toward classifying Fano fourfolds with such base loci. This work reveals a fundamental contrast with Shokurov’s dimension-three picture and supplies tools for exploring the birational geometry of high-dimensional Fano varieties with sizable anticanonical base loci.

Abstract

A famous theorem of Shokurov states that a general anticanonical divisor of a smooth Fano threefold is a smooth K3 surface. This is quite surprising since there are several examples where the base locus of the anticanonical system has codimension two. In this paper we show that for four-dimensional Fano manifolds the behaviour is completely opposite: if the base locus is a normal surface, hence has codimension two, all the anticanonical divisors are singular.

Theorems & Definitions (34)

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• proof : Proof of Proposition \ref{['proposition-exceptional-cases-B']}