Paper

{
  "raw_xml": "<entry>\n    <id>http://arxiv.org/abs/2603.19215v1</id>\n    <title>$R$-equivalence on Cubic Surfaces I: Existing Cases with Non-Trivial Universal Equivalence</title>\n    <updated>2026-03-19T17:57:38Z</updated>\n    <link href='https://arxiv.org/abs/2603.19215v1' rel='alternate' type='text/html'/>\n    <link href='https://arxiv.org/pdf/2603.19215v1' rel='related' title='pdf' type='application/pdf'/>\n    <summary>Let $V$ be a smooth cubic surface over a $p$-adic field $k$ with good reduction. Swinnerton-Dyer (1981) proved that $R$-equivalence is trivial on $V(k)$ except perhaps if $V$ is one of three special types--those whose $R$-equivalence he could not bound by proving the universal (admissible) equivalence is trivial. We consider all surfaces $V$ currently known to have non-trivial universal equivalence. Beyond being intractable to Swinnerton-Dyer's approach, we observe that if these surfaces also had non-trivial $R$-equivalence, they would contradict Colliot-Thélène and Sansuc's conjecture regarding the $k$-rationality of universal torsors for geometrically rational surfaces.\n  By devising new methods to study $R$-equivalence, we prove that for 2-adic surfaces with all-Eckardt reductions (the third special type, which contains every existing case of non-trivial universal equivalence), $R$-equivalence is trivial or of exponent 2. For the explicit cases, we confirm triviality: the diagonal cubic $X^3+Y^3+Z^3+ζ_3 T^3=0$ over $\\mathbb{Q}_2(ζ_3)$--answering a long-standing question of Manin's (Cubic Forms, 1972)--and the cubic with universal equivalence of exponent 2 (Kanevsky, 1982).\n  This is the first in a series of works derived from a year of interactions with generative AI models such as AlphaEvolve and Gemini 3 Deep Think, with the latter proving many of our lemmas. We disclose the timeline and nature of their use towards this paper, and describe our broader AI-assisted research program in a companion report (in preparation).</summary>\n    <category scheme='http://arxiv.org/schemas/atom' term='math.AG'/>\n    <category scheme='http://arxiv.org/schemas/atom' term='cs.AI'/>\n    <category scheme='http://arxiv.org/schemas/atom' term='cs.HC'/>\n    <category scheme='http://arxiv.org/schemas/atom' term='math.NT'/>\n    <published>2026-03-19T17:57:38Z</published>\n    <arxiv:comment>23 pages</arxiv:comment>\n    <arxiv:primary_category term='math.AG'/>\n    <author>\n      <name>Dimitri Kanevsky</name>\n    </author>\n    <author>\n      <name>Julian Salazar</name>\n    </author>\n    <author>\n      <name>Matt Harvey</name>\n    </author>\n  </entry>"
}