Zero-cycles on surfaces dominated by products of hyperelliptic curves

Jean-Louis Colliot-Thélène; Federico Scavia; Alexei Skorobogatov

Zero-cycles on surfaces dominated by products of hyperelliptic curves

Algebraic Geometry 2026-07-21 v1 Number Theory

Authors: Jean-Louis Colliot-Thélène , Federico Scavia , Alexei Skorobogatov

Abstract

Conditionally on the finiteness of the relevant Tate-Shafarevich groups, we prove a local-to-global result for zero-cycles of degree $1$ on the surfaces given by $y^2 = f_1(x_1)f_2(x_2)$ , where the polynomials $f_1$ and $f_2$ are algebraically general. The proof combines the fibration method, parity results for $2$ -Selmer groups in quadratic twist families, and a variant of a theorem of Morgan on the variation of the Cassels-Tate pairing.

Keywords

zero-cycles surfaces hyperelliptic curves local-to-global fibration method Selmer groups Cassels-Tate pairing

Cite

@article{arxiv.2607.18906,
  title  = {Zero-cycles on surfaces dominated by products of hyperelliptic curves},
  author = {Jean-Louis Colliot-Thélène and Federico Scavia and Alexei Skorobogatov},
  journal= {arXiv preprint arXiv:2607.18906},
  year   = {2026}
}