1 · The essentials
The manuscript challenges a proposed complete list of mathematical relations associated with curves and their marked points. It describes a relation that holds but is not covered by Pixton’s original list. The example concerns curves of genus with marked points. The relation becomes zero in both of the mathematical settings considered, yet remains outside the reach of the original relations. The manuscript presents this as a counterexample to both corresponding versions of the completeness conjecture. The claim concerns Pixton’s original system, not every possible extension of it. The main result is not formalized in Lean and has not been formally verified.
A picture to help (an imperfect image, not to be taken literally)
As an imperfect image, imagine a collection of accounting rules claimed to explain every way a balance can come out to zero. The manuscript describes a balance that really is zero but cannot be accounted for by that particular collection of rules.
Why it matters
A completeness conjecture asks whether a proposed system captures all relations of the relevant kind. The counterexample claimed here would mean that Pixton’s original relation system misses a relation in both rational Chow and rational cohomology. Its significance is this failure of completeness, rather than a failure of the original relations themselves.
2 · Going further
The precise issue is completeness of Pixton’s original relation system on a moduli space of stable pointed curves. The abstract states that the manuscript constructs a formal tautological class outside the original Pixton relation span, while its image is zero in the Chow ring and in rational cohomology. The example has genus and markings.
The key distinction is between belonging to the original relation span and vanishing after passing to the target mathematical setting. The claimed construction separates these two conditions: the formal class lies outside the span but nevertheless maps to zero. This is the mechanism described in the abstract for a counterexample to the Chow and rational-cohomological forms of the original completeness conjecture. The supplied text gives no further construction details or argument for either condition, and it supplies no companion manuscripts.
Useful notions
- Curves, genus, and marked points
- Moduli spaces of stable pointed curves
- Formal tautological classes and relation spans
- Rational Chow rings and rational cohomology
What was known before / what is new
Before
The source identifies Pixton’s original relation system and its completeness conjecture as the setting for the result. It distinguishes Chow and rational-cohomological forms of that conjecture, but gives no further history or account of earlier results.
New
The manuscript claims a formal tautological class that lies outside Pixton’s original relation span but vanishes in rational Chow, hence also in rational cohomology. The stated example uses genus and markings. It is presented as a counterexample to both forms of the original completeness conjecture.
Limits and assumptions
The stated construction concerns a moduli space of stable pointed curves at the specified genus and number of markings, with rational Chow and rational cohomology as the target settings. Its scope is Pixton’s original relation system; the supplied text does not address modified systems or smaller examples. The abstract does not provide the construction or its supporting argument, and it does not identify further open problems. The main result is not formalized in Lean and has not been formally verified.
Mini glossary
- Completeness conjecture —
- Here, the assertion that Pixton’s original relation system captures all the relations under consideration.
- Original Pixton relation span —
- The collection of formal relations obtainable by taking linear combinations of Pixton’s original relations.
- Formal tautological class —
- The kind of formal mathematical object constructed in the manuscript, whose image can be considered in Chow and cohomology.
- Vanishing —
- Having image equal to zero in the specified mathematical setting.
Keywords : Pixton completeness · Tautological relations · Rational Chow
AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.
3 · Technical details and sources
Claim made in the manuscript (source text)
Constructs a tautological relation on a moduli space of stable pointed curves that vanishes in rational Chow, hence in rational cohomology, but lies outside Pixton's original relation span. This disproves the Chow and rational-cohomological forms of his original completeness conjecture.
To be verified: the proof is written in the manuscript, but no software has checked it and no validation by independent mathematicians is established. It may contain errors.
Manuscripts in this family
A high-arity counterexample to Pixton completeness in Chow
24/09/2026We disprove the Chow and rational-cohomological forms of Pixton's completeness conjecture for his original relation system. We construct a formal tautological class outside the original Pixton relation span whose image is zero in the Chow ring and in rational cohomology. The example has genus and markings.