The geometric exceptional set for Fano threefolds

Abstract: We study the geometric exceptional set in Manin's conjecture for smooth Fano threefolds. We prove that every adjoint rigid surface with $a$-invariant one on a non-product smooth Fano threefold is normal. This yields a description of the geometric exceptional set in terms of the higher-$a$ locus, anticanonical conics, and del Pezzo fibrations. Zariski density arise only from the latter two contributions. For del Pezzo fibrations, we construct a finite collection of parameter covers which controlling the associated breaking maps. We classify their low-genus components for general smooth Fano threefolds yielding examples with potentially Zariski dense geometric exceptional sets in $22$ of the $105$ deformation families. Extending Batyrev-Tschinkel's example, we construct Fano fourfolds with quartic del Pezzo fibrations and an explicit Fano threefold with two degree-six del Pezzo fibrations, all with Zariski dense geometric exceptional sets. For this threefold, the original form of Manin's conjecture fails on every non-empty open subset for every anticanonical height.