Manin's conjecture for certain biprojective hypersurfaces

Abstract

Abstract. Using the circle method, we count integer points on complete intersections in biprojective space in boxes of different side length, provided the number of variables is large enough depending on the degree of the defining equations and certain loci related to the singular locus. Having established these asymptotics we deduce asymptotic formulas for rational points on such varieties with respect to the anticanonical height function. In particular, we establish a conjecture of Manin for certain smooth hypersurfaces in biprojective space of sufficiently large dimension.

Density of Rational Points on a Certain Smooth Bihomogeneous Threefold

“…The circle method is a traditional technique to count solutions to diophantine equations, and it has recently been applied by Schindler [9,10] to count rational points on bihomogeneous varieties. In particular, [10,Theorem 1.2] states that smooth hypersurfaces in biprojective space P n 1 × P n 2 defined by general bihomogeneous forms…”

“…In particular, [10,Theorem 1.2] states that smooth hypersurfaces in biprojective space P n 1 × P n 2 defined by general bihomogeneous forms…”

Counting rational points on biquadratic hypersurfaces

“…By way of comparison, Schindler's result [13] yields the same conclusion for n > 193. (In fact the latter result also handles complete intersections in P n 1 −1 × P n 2 −1 when min{n 1 , n 2 } is large enough.…”

“…This has been extended to n 3 by Spencer [14]. For general bidegree (d 1 , d 2 ) the best result available is due to Schindler [13], who has verified the Manin-Peyre conjecture (1.1) for an appropriate open subset U ⊂ X, provided that n > 3 · 2 d 1 +d 2 d 1 d 2 + 1.…”

“…(Here we have used [13,Lemma 2.2] to note that 2n − max{dim V * 1 , dim V * 2 } n − 1 in [13, Thm. 1.1].…”