Counting Rational Points on the Stacky $\operatorname{Sym}^2\mathbb{P}^1$

TL;DR

This paper proves a weak form of a conjecture related to counting rational points on a specific algebraic stack, demonstrating that certain substructures dominate the distribution of these points.

Contribution

It establishes the weak form of the generalized Batyrev-Manin-Malle conjecture for the symmetric square of the projective line, highlighting the accumulation on the diagonal substack.

Findings

Diagonal substack is an accumulating set of rational points.

Weak form of the Batyrev-Manin-Malle conjecture verified for the stack.

Provides insight into the distribution of rational points on algebraic stacks.

Abstract

We prove the weak form of the generalized Batyrev-Manin-Malle conjecture formulated in \cite{ellenberg2021heights} for the stack $\operatorname{Sym}^2\mathbb{P}^1 := (\mathbb{P}^1 \times \mathbb{P}^1)/S_2$, where the $S_2$ action just permutes the two coordinates. In particular, we show that the diagonal $\Delta \subset \operatorname{Sym}^2\mathbb{P}^1$ is an accumulating substack.