发表机构
University of Georgia; Leibniz University Hannover（佐治亚大学; 汉诺威莱布尼茨大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文给出Cassels--Swinnerton-Dyer猜想在三次曲面情形（含光滑与奇异）的完整证明，通过两则独立证明补全缺失的4次点情形，并利用Ma的提升论证推广至任意域。
AI 中文摘要
Cassels--Swinnerton-Dyer猜想断言：一个三次超曲面包含一个有理点当且仅当它包含一个与$3$互素的次数点，或等价地，包含一个次数为$1$的零环。特征零中光滑三次曲面的情形已被Coray和Voisin归约到次数为$4$的点的情况。我们给出这一缺失情形的两则独立证明，并利用Ma的提升论证将结果推广到任意域上的光滑三次曲面。我们进一步对奇异三次曲面的情形给出一个单独的论证，扩展了Coray在完美域上的先前工作。总之，这证明了三次曲面的Cassels--Swinnerton-Dyer猜想。
英文摘要
The Cassels--Swinnerton-Dyer conjecture asserts that a cubic hypersurface contains a rational point if and only if it contains a point of degree coprime to $3$, or, equivalently, a zero-cycle of degree $1$. The case of smooth cubic surfaces in characteristic zero has been reduced by Coray (1976) and Voisin (2026) to the case of points of degree $4$. We give two independent proofs of this missing case and use a lifting argument to extend the result to smooth cubic surfaces over arbitrary fields. We further give a separate argument for the case of singular cubic surfaces, extending previous work of Coray over perfect fields. Altogether, this proves the Cassels--Swinnerton-Dyer conjecture for cubic surfaces.
Comments16 pages; v2: Details added to AI disclosure; v3: minor changes
