PreprintCC BY-NC-ND 4.0arXiv
Counting Rational Points on a Certain Bihomogeneous Threefold
Abstract
Let V⊂ℙ2×ℙ2 be the threefold x0y02+x1y12+x2y22=0, and let U be the complement of its coordinate hyperplane sections. For the anticanonical height H([x],[y])=|x|2|y|, evaluated on primitive integral representatives, we prove NU(B)=(12π−14)/ζ(2)ζ(3)Blog B+o(Blog B). The two projections combine lattice counting with uniform conic estimates. The main arithmetic step is an average of the fiber singular series, whose two main terms are related by Hilbert reciprocity. A moment estimate controls the full bad-prime part, and shape estimates permit the errors to be summed over the height region. We also check the leading constant with Peyre's constant.
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BibTeX
@misc{zhao2026counting,
title = {{Counting Rational Points on a Certain Bihomogeneous Threefold}},
author = {Xiaodong Zhao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.15315},
eprint = {2610.15315},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.15315},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Counting Rational Points on a Certain Bihomogeneous Threefold
AU - Xiaodong Zhao
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.15315
UR - https://arxiv.org/abs/2610.15315
AB - Let V⊂ℙ^2×ℙ^2 be the threefold x_0y_0^2+x_1y_1^2+x_2y_2^2=0, and let U be the complement of its coordinate hyperplane sections. For the anticanonical height H([x],[y])=|x|^2|y|, evaluated on primitive integral representatives, we prove N_U(B)=(12π−14)/ζ(2)ζ(3)Blog B+o(Blog B). The two projections combine lattice counting with uniform conic estimates. The main arithmetic step is an average of the fiber singular series, whose two main terms are related by Hilbert reciprocity. A moment estimate controls the full bad-prime part, and shape estimates permit the errors to be summed over the height region. We also check the leading constant with Peyre's constant.
ER -