A Zariski dense exceptional set in Manin's conjecture: dimension 2

TL;DR

This paper constructs a specific del Pezzo surface of degree 1 with a Zariski dense geometric exceptional set, providing the first counterexample to Manin's Conjecture in dimension 2 over characteristic 0, assuming certain finiteness conjectures.

Contribution

It presents the first explicit example of a del Pezzo surface with a Zariski dense exceptional set, challenging the original formulation of Manin's Conjecture in dimension 2.

Findings

Constructed a degree 1 del Pezzo surface with dense exceptional set

Provided the first counterexample to Manin's Conjecture in dimension 2

Linked the existence of counterexamples to finiteness of Tate-Shafarevich groups

Abstract

Recently, Lehmann, Sengupta, and Tanimoto proposed a conjectural construction of the exceptional set in Manin's Conjecture, which we call the geometric exceptional set. We construct a del Pezzo surface of degree $1$ whose geometric exceptional set is Zariski dense. In particular, this provides the first counterexample to the original version of Manin's Conjecture in dimension $2$ in characteristic $0$. Assuming the finiteness of Tate-Shafarevich groups of elliptic curves over $\mathbb{Q}$ with $j$-invariant $0$, we show that there are infinitely many such counterexamples.