Local solubility of a family of ternary conics over a biprojective base I

TL;DR

This paper investigates the local solubility of a family of ternary conics over a biprojective base, providing bounds on the number of rational points and conditions for their existence.

Contribution

It offers new upper bounds on rational points for a family of ternary conics and identifies conditions for their lower bounds, advancing understanding of their local solubility.

Findings

Upper bounds for the number of rational points on the conics.

Conditions under which lower bounds for rational points exist.

Analysis of local solubility over biprojective bases.

Abstract

Let $f,g\in\mathbb{Z}[u_1,u_2]$ be binary quadratic forms. We provide upper bounds for the number of rational points $(u,v)\in\mathbb{P}^1(\mathbb{Q})\times\mathbb{P}^1(\mathbb{Q})$ such that the ternary conic \[ X_{(u,v)}: f(u_1,u_2)x^2 + g(v_1,v_2)y^2 = z^2 \] has a rational point. We also give some conditions under which lower bounds exist.