Breakthrough Tracker record
Two internal proof routes claim the Cassels–Swinnerton-Dyer criterion for cubic surfaces
Alexeev and Schreieder claim that a cubic surface over any field has a rational point exactly when it has a point of degree coprime to three, equivalently a zero-cycle of degree one. Their paper gives two proofs of the missing degree-four case and smooth and singular extensions.
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- Stable ID
math-cassels-swinnerton-dyer-cubic-surfaces-2026- Revision
math-cassels-swinnerton-dyer-cubic-surfaces-2026.v1- Field
- Mathematics · Arithmetic and algebraic geometry
- Evidence
- Tier 1 · Peer reviewed: No
- Record state
- Provisional · Provisional dual-proof resolution
- Last checked
AI role
The authors describe independently GPT-assisted proof development: GPT-6 Astra supplied a rational-section idea and extensions for one route, while GPT assistance contributed to the second. The authors synthesized and checked the arguments.
Record details
- Problem or result
- The rational-point criterion for cubic surfaces
- Authors
- Valery Alexeev and Stefan Schreieder
- Institutions
- University of Georgia; Leibniz University Hannover
- Result date
- Submitted September 14; current v3 September 25, 2026
Why it matters
It would close the cubic-surface case of a longstanding rational-point criterion across characteristics and singularity cases.
Limits
The two routes are within one coauthored paper, not independent external verification. Expert comments are acknowledged, but the preprint is not peer reviewed.
Sources
Correction and revision history
- 2026-10-02 — Added after primary-source, scope, status, AI-role and limitation review.
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