论文标题
Manin的猜想中的Zariski密集特殊设置：维度2
A Zariski dense exceptional set in Manin's conjecture: dimension 2
论文作者
论文摘要
最近，莱曼（Lehmann），Sengupta和Tanimoto提出了Manin猜想中特殊场景的猜想结构，我们称之为几何特殊集合。我们构建了$ 1 $的Del Pezzo表面，其几何特殊集合是Zariski密集。特别是，这提供了对Manin猜想的原始版本的第一个反例，以$ 2 $的特征$ 0 $。假设泰特 - 夏法尔维奇（Tate-Shafarevich）的有限属于$ \ mathbb {q} $，$ j $ -invariant $ 0 $，我们表明有许多这样的反例。
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.
