Classification of Fano fourfolds with large anticanonical base locus

TL;DR

This paper classifies smooth Fano fourfolds with a smooth surface as the base of their anticanonical system, revealing exactly 22 deformation families connected to Fano threefolds with a P^1-bundle structure.

Contribution

It provides a complete classification of these fourfolds and shows they are all constructed via a unified geometric method.

Findings

Exactly 22 deformation families identified

All families derived from a common geometric construction

Connected to Fano threefolds with P^1-bundle structures

Abstract

We give a classification of smooth Fano fourfolds such that the base scheme of the anticanonical system is a smooth surface. As a consequence we show that there are exactly 22 deformation families of such manifolds and they are all obtained by the same geometric construction. These 22 families are closely related to the list of smooth Fano threefolds that admit a $\mathbb P^1$-bundle structure.