Zero-cycles on surfaces dominated by products of hyperelliptic curves

2026/07/21 by Jean-Louis Colliot-Thélène, Federico Scavia, Alexei Skorobogatov
#math.AG #math.NT

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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 y² = f₁(x₁)f₂(x₂), where the polynomials f₁ and f₂ 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.

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