Counting points on Hilb^mℙ² over function fields

2019/05/12 by Adelina Mânzăţeanu, Mânzăţeanu, Adelina
Mathematics · #11G35 (14C05 #14G05) #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Meromorphic and Entire Functions #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1905.04772

openalex publication_date 2019/05/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 Hilb²ℙ² 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 = ℙ².

Citations

• Rational points of bounded height on Fano varieties
• Sur le nombre des points rationnels de hauteur borné des variétés algébriques
• Algebraic Families on an Algebraic Surface

Related

• Sharp bounds for the number of rational points on algebraic curves and dimension growth, over all global fields
• What is the height of two points in the plane?
• Rational points of bounded height and the Weil restriction
• Counting integral points of bounded height on varieties with large fundamental group
• Counting points on Hessenberg Varieties over finite fields