Classification of Fano fourfolds with large anticanonical base locus
Andreas Höring, Saverio Andrea Secci
TL;DR
The work classifies smooth Fano fourfolds X whose anticanonical base locus |-K_X| is a smooth surface, establishing 22 deformation families. The authors build a uniform construction: blowing up the base B to obtain X', which carries a flat Weierstrass elliptic fibration with distinguished section E ≅ P(N^*_{B/X}), and they show B and N^*_{B/X}^ are Fano; the fourfold X arises from the data (B, N^*_{B/X}) via a Weierstrass/elliptic framework. A key structural result identifies a π: X → V with del Pezzo degree-one fibers and relates Bs(|-K_X|) to a π-section, while X' → T is a flat elliptic fibration compatible with the blow-up. The classification reduces to analyzing the base V (P^2 or P^1 × P^1) and the rank-two Fano bundles N^*_{B/X} over V, yielding 22 families corresponding to Fano threefolds with a P^1-bundle structure (SW90). The work provides explicit constructions (Examples) and assembles a final invariant table, highlighting the geometric regularity across the families and contributing a comprehensive link between fourfolds and their threefold progenitors.
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.