Density of rational points on a quadric bundle in $\mathbb{P}^3\times \mathbb{P}^3$

Abstract

An asymptotic formula is established for the number of rational points of bounded anticanonical height which lie on a certain Zariski dense subset of the biprojective hypersurface= 0 in P 3 × P 3 . This confirms the modified Manin conjecture for this variety, in which the removal of a "thin" set of rational points is allowed.prim of F (x; y) = 0 in the region |x| X, |y| Y , such that ∆(x) = . Similarly, we writeThe primary result in this section is the following collection of upper bounds.Lemma 2.1. We havefor any ε > 0.

Weak approximation and the Hilbert property for Campana points

“…Thin sets play an important role in modern refinements of Manin's conjecture originally due to Peyre [19], in which the exceptional set is thin (see [17,Thm. 1.3] and [6,Thm. 1.1] for examples and [16, §5] for an overview).…”

Counting rational points on biquadratic hypersurfaces

“…We now know that (1.1) is false in general. Recent work of Browning and Heath-Brown [2] shows that for the smooth hypersurface {x 1 y 2 1 + • • • + x 4 y 2 4 = 0} ⊂ P 3 × P 3 , of bidegree (1,2), one needs to remove certain "thin sets" of rational points in order to arrive at the Manin-Peyre prediction. For larger values of n, however, we expect that it suffices to merely remove proper closed subvarieties, as allowed for in (1.1).…”

“…Our proof of Theorem 1.1 uses the Hardy-Littlewood circle method and draws heavily on the strategies adopted in the works of Browning-Heath-Brown [2] and Schindler [13] that we have already described. The main idea is to begin by proving an asymptotic formula for the number of (restricted) integer solutions (x, y) ∈ Z n prim × Z n prim to the equation F (x; y) = 0, with |x| X and |y| Y , for arbitrary X, Y 1.…”

Geometric aspects of Manin's Conjecture

“…However, there are also many counterexamples to this version of Manin's Conjecture first found by Batyrev and Tschinkel in [BT96b]. (See, e.g., [LR14], [BL17], and [BHB18] for other counterexamples to the closed set version of Manin's Conjecture.) In [Pey03], Peyre first predicted that an exceptional set in Manin's Conjecture should be contained in a thin subset and so far there is no counterexample to this version of Manin's Conjecture.…”

“…Proving Conjecture 1.2 for (X, L) is out of reach at this moement. (Though the Fano fivefold defined by i x i y 2 i = 0 has been handled in [BHB18].) 2.2.…”