arXiv subjects

Runxuan Gao

Publications and source records attributed to Runxuan Gao.

3 recordsLinked to original sources

arXiv2024

Quasi-\'etale covers of Du Val del Pezzo surfaces and Zariski dense exceptional sets in Manin's conjecture

We construct first examples of singular del Pezzo surfaces with Zariski dense exceptional sets in Manin's conjecture, varying in degrees $1, 2$ and $3$. The obstructions arise from accumulating quasi-\'etale covers. We classify all quasi-\'etale covers of Du Val del Pezzo surfaces, extending earlier works of Miyanishi-Zhang. Then, we identify all potential examples by studying group actions on the pseudo-effective cones, and show that no such example exists in degree more than $3$. Relevant results on the geometry and descent problem of quasi-\'etale covers are also established, providing a systematic method to construct other examples.

math.AG↗

arXiv2023

The geometric exceptional set in Manin's conjecture for Batyrev and Tschinkel's example

Batyrev and Tschinkel's example is a Fermat cubic surface bundle $X$ which is a Fano $5$-fold. It is the first example for which Manin's conjecture can never hold for a proper closed exceptional set. Recently, Lehmann, Sengupta, and Tanimoto proposed a conjectural geometric description of the exceptional set in Manin's conjecture and showed that it is always contained in a thin set. Over a field of characteristic $0$, we explicitly construct finitely many thin maps such that any thin map $f:Y\rightarrow X$ with equal or larger $a$- and $b$-values in lexicographical order factors rationally through one of them. In particular, this defines a thin set which coincides with Lehmann-Sengupta-Tanimoto's conjectural exceptional set.

math.AG↗

arXiv2023

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

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.

math.AG↗