Fano fourfolds with large anticanonical base locus

TL;DR

This paper explores the behavior of anticanonical divisors in four-dimensional Fano manifolds, revealing that a normal surface base locus leads to all anticanonical divisors being singular, contrasting with the threefold case.

Contribution

It establishes a stark contrast between threefold and fourfold Fano manifolds regarding the singularity of anticanonical divisors based on their base locus.

Findings

In fourfold Fano manifolds, a normal surface base locus implies all anticanonical divisors are singular.

This behavior is opposite to the threefold case where general anticanonical divisors are smooth K3 surfaces.

The result highlights a fundamental difference in the geometry of Fano manifolds across dimensions.

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.