[Paper Review] Counting points on $ ext{Hilb}^m\mathbb{P}^2$ over function fields

Adelina Mânzăţeanu|arXiv (Cornell University)|May 12, 2019

Algebraic Geometry and Number Theory20 references4 citations

TL;DR

This paper establishes an asymptotic formula for the number of $K$-rational points of bounded height on the Hilbert scheme $\operatorname{Hilb}^2\mathbb{P}^2$ over a global function field $K$ of positive characteristic, showing that after removing a thin set, the leading constant matches Peyre's conjectural prediction. It extends the analogy between $0$-cycles over finite fields and number fields, proving a function field analogue of the prime number theorem for $\mathbb{P}^2$. The result supports the refined version of Manin’s conjecture in positive characteristic.

ABSTRACT

Let $K$ be a global field of positive characteristic. We give an asymptotic formula for the number of $K$-points of bounded height on the Hilbert scheme $ ext{Hilb}^2\mathbb{P}^2$ and show that by eliminating an exceptional thin set, the constant in front of the main term agrees with the prediction of Peyre in the function field setting. Moreover, we extend the analogy between the integers and $0$-cycles on a variety $V$ over a finite field to $0$-cycles on a variety $V$ over $K$ and establish a version of the prime number theorem in the case when $V = \mathbb{P}^2$.

Motivation & Objective

• To verify the refined version of Manin’s conjecture for $\operatorname{Hilb}^2\mathbb{P}^2$ over global fields of positive characteristic by establishing an asymptotic formula for $K$-rational points of bounded height.
• To demonstrate that the leading constant in the asymptotic formula agrees with Peyre’s conjectural prediction in the function field setting, after removing a thin set of exceptional points.
• To extend the analogy between $0$-cycles on varieties over finite fields and number fields, proving a function field analogue of the prime number theorem for $\mathbb{P}^2$.
• To generalize counting techniques from number fields to function fields, using height zeta functions and geometric decomposition of the Hilbert scheme.

Proposed method

• The method begins by decomposing $\operatorname{Hilb}^2\mathbb{P}^2(K)$ into geometric strata based on the structure of $0$-cycles, distinguishing between irreducible and reducible configurations.
• It uses a height function $H_{\omega^{-1}}$ associated with the anticanonical line bundle to count points of bounded height $q^M$, reducing the problem to counting points in $\mathbb{P}^2$ and symmetric powers over $K$.
• The proof relies on a function field analogue of Schmidt’s and Schanuel’s counting results for points of bounded degree, with height defined via the adelic product formula over places of $K$.
• It applies asymptotic estimates for $N_K(n+1,d,M)$, the number of points of degree $d$ over $K$ with height $q^{M/ed}$, using zeta functions and class numbers of function fields.
• The main contribution is derived by analyzing contributions from reducible configurations: products of $m-2k$ points on $\mathbb{P}^2$ and $k$ copies of $\operatorname{Sym}^2\mathbb{P}^2$, with the dominant term arising when $k=1$.
• A key technical step involves showing that higher-order terms (e.g., $k>1$) contribute only $O(M^{m-4})$, while the $k=1$ case yields $\sim c q^M M$ with $c$ matching Peyre’s constant.

Experimental results

Research questions

• RQ1 Does the leading constant in the asymptotic count of $K$-rational points on $\operatorname{Hilb}^2\mathbb{P}^2$ match Peyre’s conjectural prediction in the function field setting?
• RQ2 Can the analogy between $0$-cycles over finite fields and number fields be extended to global function fields, particularly in the context of height zeta functions?
• RQ3 What is the precise asymptotic behavior of the number of $K$-points of bounded height on $\operatorname{Hilb}^2\mathbb{P}^2$ over a function field $K$ of positive characteristic?
• RQ4 How do reducible configurations contribute to the point count, and can higher-order reducible terms be shown to be lower-order compared to the main term?

Key findings

• The number of $K$-rational points of height $q^M$ on $\operatorname{Hilb}^2\mathbb{P}^2$, after removing a thin set $Z_0$, is asymptotically $c q^M M + O(\sqrt{M} q^M)$ as $M \to \infty$.
• The leading constant $c$ in the asymptotic formula matches Peyre’s conjectural constant for the function field setting, confirming the refined Manin conjecture in this case.
• The main contribution to the point count comes from reducible $0$-cycles consisting of one pair of conjugate points over $K$ and $m-2$ rational points, corresponding to $k=1$ in the decomposition.
• Contributions from configurations with $k>1$ irreducible components are bounded by $O(M^{m-4})$, which is lower order than the $M$-linear term, confirming the dominance of the $k=1$ case.
• The paper establishes a function field analogue of the prime number theorem for $\mathbb{P}^2$, where the number of $K$-rational points of bounded height grows like $q^M M$ with a constant related to zeta functions and class numbers.
• The analysis confirms that the $M^{m-2}$ terms cancel in the asymptotic expansion, leading to a main term of order $M^{m-3}$, which is consistent with the expected growth in the refined Manin conjecture.