diff 9506729..56bc728 pixton_completeness_counterexample_v2.txt
diff --git a/pixton_completeness_counterexample_v2.txt b/pixton_completeness_counterexample_v2.txt
index f119aad..063d7d4 100644
--- a/pixton_completeness_counterexample_v2.txt
+++ b/pixton_completeness_counterexample_v2.txt
@@ -5,156 +5,6 @@ PIXTON COMPLETENESS: A COUNTEREXAMPLE
-=====================================================
-PART 0: WHY YOU MIGHT CARE
-=====================================================
-
-Here is the whole paper in one sentence:
-
- Mathematicians guessed a complete set of rules for measuring
- curves. We found one measurement their rules cannot explain --
- and yet that measurement still comes out equal to zero when we
- actually compute it.
-
-The rest of Part 0 unpacks that sentence.
-
--------------------------------------------------------------------------
-0.1 What is a curve, in math?
--------------------------------------------------------------------------
-
-A curve, in the algebraic-geometric sense used in this paper, is a
-shape that locally looks like the real line (a real curve) or the
-complex plane (a complex curve). Key examples:
-
- - a circle;
- - a sphere, which is a real surface (dimension 2 over R);
- - a torus, which is the surface of a donut;
- - an n-holed donut, for any n >= 0.
-
-"Genus" is the number of holes. A sphere has genus 0, a torus has
-genus 1, a two-holed donut has genus 2, and so on.
-
-A complex curve is a surface where we additionally know how to add
-and multiply points (in a way compatible with the shape). This paper
-works with smooth compact complex curves of a fixed genus g, plus a
-finite set of "marked points" (distinguished labeled locations on
-the curve, numbered 1 through n).
-
--------------------------------------------------------------------------
-0.2 What is "the space of all curves of genus g"?
--------------------------------------------------------------------------
-
-Take all curves of a fixed shape -- say, all genus-g curves with n
-marked points. This is not one curve; it is a family of curves that
-itself varies. In fact this family has the structure of a shape (a
-"space") in its own right: it is called the moduli space and is
-written M_{g,n}.
-
-A point in M_{g,n} is "one curve-with-markings." The whole space
-M_{g,n} is all curves packed together, roughly like a rainbow of
-every possible genus-g shape smoothly varying between its neighbors.
-
-M_{g,n} is not compact -- you can let a curve "degenerate" by
-pinching a bit of it down to a point. That produces a boundary;
-adding the boundary produces the compact moduli space of stable
-curves, Mbar_{g,n}. Stable curves are the curves that appear at
-the boundary: ordinary smooth curves plus curves with a limited
-kind of singularity called a node.
-
--------------------------------------------------------------------------
-0.3 What can we measure on a curve?
--------------------------------------------------------------------------
-
-At every marked point i there is a natural line bundle of tangent
-directions, whose first Chern class is written psi_i. There are
-also kappa classes kappa_j that measure how the marked points fall
-around a curve.
-
-The important point: psi and kappa are canonical, universal
-measurements. Every smooth curve has them; they vary continuously
-as the curve varies through moduli.
-
-Beyond psi and kappa, we can build more measurements by multiplying
-psi's and kappa's, and by pushing forward along forgetful maps
-(forget a marked point) and gluing maps (glue two marked points,
-creating a node). All measurements built this way are called
-tautological classes.
-
-The set of all tautological classes is a ring called the
-tautological ring, written R^*(Mbar_{g,n}).
-
--------------------------------------------------------------------------
-0.4 What is a "relation"?
--------------------------------------------------------------------------
-
-A relation in the tautological ring is a rule that says: "whenever
-you compute these particular measurements combinatorially, they add
-up to zero."
-
-Equivalently, a relation is a formal tautological class that
-actually evaluates to zero when you test it against every cycle.
-
-Two things must be distinguished carefully:
-
- 1. A formal combination of measurements. This lives in the
- strata algebra S_{g,n} -- a giant free abelian group on formal
- symbols. It has no geometry inside it; it is pure bookkeeping.
-
- 2. The realization map q: S_{g,n} -> A^*(Mbar_{g,n}),
- which substitutes the actual psi and kappa classes for their
- formal symbols.
-
-A proposal for "a complete list of relations" is a proposal for
-what q's kernel is: the set of formal combinations that always
-evaluate to zero.
-
--------------------------------------------------------------------------
-0.5 What did Pixton propose?
--------------------------------------------------------------------------
-
-In 2012 Aaron Pixton proposed a large explicit family of formal
-relations. His relations look like sums over stable graphs,
-decorated with specific generating series. He guessed that the only
-tautological relations were the ones his rules produce, possibly
-multiplied and moved around by formal gluing operations.
-
-Concretely: let P_{g,n} be the formal span of all Pixton relations
-plus all their formal consequences. Pixton's conjecture (for the
-original system) is:
-
- ker q = P_{g,n}
-
-for every stable pair (g,n).
-
-In words: every formal combination that vanishes in actual Chow
-theory should be expressible using Pixton's rules plus gluing.
-
--------------------------------------------------------------------------
-0.6 What this paper proves
--------------------------------------------------------------------------
-
-This paper constructs a stable pair (g,n) with g = 10^60 and
-n = 3 * C(g,3), and a specific formal class Y in S_{g,n}, with
-two properties:
-
- (a) q(Y) = 0 in actual Chow theory -- so Y is a real relation,
- not a formal wish;
- (b) Y is NOT in Pixton's span P_{g,n}.
-
-Together, (a) and (b) say:
-
- ker q_{g,n} is STRICTLY BIGGER than P_{g,n}.
-
-Pixton's conjecture, in its "original system" form, is false.
-
-Because cycle class commutes with everything in sight, part (b)
-also shows the analogous statement in rational cohomology: Y is
-a cohomological relation that Pixton's list does not generate.
-
-That is what the paper is about. Parts 1 through 6 build the
-machinery to prove (a) and (b). Part 7 discusses the context.
-
-
=====================================================
PART 1: BACKGROUND
=====================================================
@@ -861,40 +711,6 @@ reader that Y is a legitimate element rather than a formal
wish.
--------------------------------------------------------------------------------
-3.8 Roadmap: how Theorem 2.1 splits
---------------------------------------------------------------------------------
-
-Proving Theorem 2.1 splits into two completely independent
-claims, proved by completely different methods:
-
- Claim A (non-membership): Y is not in P_{g,n}.
-
- Claim B (vanishing): q_{g,n}(Y) = 0.
-
- A and B involve different algebras and different techniques:
-
- - Claim A uses a "formal detector": a Q-linear map out of
- S_{g,n} that annihilates P_{g,n} but takes nonzero values
- on Y. The detector can be constructed entirely within the
- strata algebra; it doesn't touch actual Chow geometry.
- This is Part 4.
-
- - Claim B uses a sophisticated analysis of relative Chow
- correspondences on curve powers, surface Quot schemes,
- a combinatorial identity from Hilbert schemes of points,
- and a support-dimension induction over the boundary.
- This is Part 5 (surface relation) plus Part 6
- (compactification and support).
-
-These two claims are independent. Part 4 only uses the formal
-product rule; Part 5-6 only use actual Chow geometry. It is our
-claim that both facts are true, and together they prove
-Theorem 2.1.
-
-End of Part 3. Part 4 builds the formal detector.
-
-
-=====================================================
PART 4: WHY Y IS NOT IN THE PIXTON SPAN
=====================================================
@@ -1916,43 +1732,6 @@ quotient by which we already lost information when we restricted
to the surface.
--------------------------------------------------------------------------------
-5.5 What Part 5 has given us
---------------------------------------------------------------------------------
-
-We now have:
-
- (i) A faithful model of Chow correspondences on curve powers
- in a coefficient algebra B_b with an odd symplectic W_b
- and a projected cubic phi.
-
- (ii) A surface Quot relation: T_gamma^{t_0} = 0 for
- isotropic-gamma contractions, t_0 = 25 (b-1)^2 - 1.
-
- (iii) The scalar a_{N,t_0} in the surface relation is
- nonzero.
-
- (iv) An exterior-ideal argument: arbitrary color products
- of >= D_E - d_e/2 + t_b pure cubic entries vanish
- modulo base codimension b-1.
-
-What's missing, and what Part 6 supplies:
-
- (v) The compactification step: the class Y, restricted to
- the boundary, agrees with a corrected cubic that we can
- compare to the pure cubic entries from (iv), up to
- controlled errors over a dense open.
-
- (vi) Support induction: iteratively reduce the dimension of
- the base-image of the residual loca, using (v),
- until the support is empty.
-
-Part 6 does (v) and (vi). That completes Claim B and the proof
-of Theorem 2.1.
-
-End of Part 5. Part 6 handles the compactification.
-
-
-=====================================================
PART 6: COMPACTIFICATION AND THE LAST STRETCH
=====================================================
@@ -2607,503 +2386,5 @@ End of Part 6.
=====================================================
-PART 7: WHAT THIS CHANGES
-=====================================================
-
-We've now seen the whole proof. Here: what it means in
-context, and what remains open.
-
---------------------------------------------------------------------------------
-7.1 The main consequence
---------------------------------------------------------------------------------
-
-Pixton's original 2012 conjecture asked whether his proposed
-system of stable-graph relations was COMPLETE: whether
-ker q_{g,n} = P_{g,n} for every stable (g,n). This paper shows
-the answer is no: there is at least one stable pair for which
-ker q is strictly bigger than P.
-
-The proof is a single counterexample. That's enough. Once
-completeness fails somewhere, the uniform conjecture is false.
-
-Equivalently: the family of Pixton relations, though it produces
-many true relations, is not exhausted by the ones the recipe
-gives. There is a tautological relation of a different kind,
-one that no amount of Pixton-recipes (multiplied, glued,
-forgotten) can produce.
-
-The specific witness Y is amusingly small in a certain sense:
-it is a codimension-2 D_* class, built from pairs of rational
-tail divisors and a rational-bridge correction, alternated
-against color-symmetric shuffles. Its existence was not
-suspected from any particular low-complexity example, which is
-why the conjecture lasted as long as it did.
-
---------------------------------------------------------------------------------
-7.2 Pixton's own follow-up work: non-Gorenstein at M_25
---------------------------------------------------------------------------------
-
-Completeness is a different question from the GORENSTEIN
-PROPERTY.
-
- - The Gorenstein property asks: is multiplication into
- complementary degree a nondegenerate pairing?
- - Completeness asks: are all relations among generators
- produced by Pixton's rules?
-
-These are independent questions. The Gorenstein property can
-fail while completeness holds, and vice versa.
-
-Pixton himself proved that R^*(M_25) is not Gorenstein
-(Pixton 2026, Theorem 1.1, Section 2.2), while establishing
-Faber-Zagier completeness in that genus. That is a case where
-the "nice geometric property" fails but the "list of relations
-is complete" property holds.
-
-So the two failure modes are truly different, and this paper's
-failure is of the "completeness" kind. That's important
-conceptually: you cannot use this paper to deduce anything new
-about Gorenstein-ness at M_25, or vice versa.
-
---------------------------------------------------------------------------------
-7.3 Related work
---------------------------------------------------------------------------------
-
-Canning-Larson-Schmitt (2026). Proves completeness together with
-non-Gorenstein behavior on the COMPACT-TYPE spaces
-M^ct_6, M^ct_{5,2}, M^ct_7. These are compactifications that
-include only curves of compact type (no separating nodes). The
-technique relies on the special structure of compact-type. Our
-counterexample lives on the FULL stable compactification; there,
-Pixton's rules are not complete. So there is no contradiction:
-completeness on a smaller space (compact type) does not imply
-completeness on the full Mbar.
-
-Janda's discriminant paper (2018). Compares two methods for
-producing relations from reconstructions of semisimple CohFTs
-with nonempty discriminant. That comparison identifies the
-systems arising from two different cancellation schemes.
-It does NOT identify the entire Chow kernel with Pixton's
-system. Our paper's comparison is with the specific Pixton
-system P_{g,n}; Janda's comparison is orthogonal to that.
-
-PPZ (2015) and Janda (2017). Proved validity of Pixton's
-system: P_{g,n} is a subset of ker q_{g,n}. These proofs use
-3-spin CohFT and equivariant GW theory of P^1 respectively.
-Validity is a strong form of "these are indeed relations".
-Completeness is the reverse direction, and this paper refutes it.
-
-Canning (2025). Surveys the tautological ring and its
-Gorenstein property for full Mbar and various subspaces,
-discussing exactly the "original system" being refuted here.
-That paper states the Chow and rational-cohomological
-completeness of the original Pixton system as a conjecture
-(Conjecture 1 in Canning 2025). This paper provides a
-counterexample to that specific statement.
-
---------------------------------------------------------------------------------
-7.4 Different flavors of "Pixton conjecture"
---------------------------------------------------------------------------------
-
-One caution about language. The phrase "Pixton's conjecture" is
-sometimes used to refer to different statements depending on
-the community. Three distinct versions:
-
- V1. Original Pixton 2012 graph formula gives a COMPLETE system.
-
- V2. Weaker version restricted to specific colors, specific
- decorations, or specific subspace of Mbar_{g,n}. Failure
- of V1 does NOT refute all such weak versions.
-
- V3. Complete-system conjecture only for the "operational"
- closure P_{g,n} (Pixton span including all Pixton-ideal
- consequences).
-
-This paper refutes V1 and V3, but does not directly refute
-narrower V2 variants that, e.g., restrict to compact type
-(Canning-Larson-Schmitt have positive results there) or to
-low genus (Pixton at M_25).
-
-When you see "Pixton's conjecture is false" in the popular
-press: check which version is being discussed. This paper
-specifically refutes the ORIGINAL full-system version, and
-that's what makes it non-trivial.
-
---------------------------------------------------------------------------------
-7.5 What remains open
---------------------------------------------------------------------------------
-
-Three directions remain open and interesting:
-
- (a) Other counterexamples in lower genus. The construction
- uses g = 10^60 to make the color budget and the
- support-dimension clear. There may be infinitely many
- lower-genus counterexamples, and characterizing the
- exact set of (g,n) where Pixton completeness holds is
- open. This paper only refutes the conjecture at one
- explicit (g,n).
-
- (b) Characterize ker q. This paper's Y is a specific class
- that's in ker q but not in P. What the FULL kernel is
- (i.e., how to describe an enlarged relation system that
- IS complete) remains open. What kind of relations do you
- need to add to Pixton's system to get ker q exactly?
-
- (c) Gorenstein-ness beyond M_25. Pixton proved
- non-Gorenstein at M_25. Canning-Larson-Schmitt
- completed compact-type cases. The full
- R^*(Mbar_{g,n}) Gorenstein behavior across all (g,n)
- remains under investigation. This paper doesn't touch
- that question except by example: the failure of
- completeness does NOT imply failure of Gorenstein.
-
-Beyond these three, there is a longer-term question: what is
-the right conceptual framework for the tautological ring that
-would give completeness in a broader sense? Candidates exist
-(e.g., adding cross-boundary "Mumford-type" relations, or
-working in a quotient that naturally produces the correct
-kernel). This paper's Y is meant to be a stepping stone:
-it tells you the original system was too small, and the proof
-techniques (exterior-ring detectors, surface Quot relations,
-support induction) may generalize to a full description.
-
---------------------------------------------------------------------------------
-7.6 Summary of what just happened
---------------------------------------------------------------------------------
-
-To recap, in one paragraph: we built an explicit class Y in
-the strata algebra S_{g,n} for g = 10^60 and n = 3*C(g,3). We
-proved two independent facts about Y:
-
- (i) Y is not in the Pixton span P_{g,n}. (Part 4: formal
- detector over an odd Frobenius algebra.)
-
- (ii) Y vanishes in rational Chow. (Parts 5-6: surface
- Quot relation plus compactification plus support
- induction.)
-
-Together, (i) and (ii) say ker q_{g,n} is strictly bigger than
-P_{g,n}. By the cycle-class map, the same failure holds in
-rational cohomology. This refutes Pixton's original
-completeness conjecture.
-
-The two proofs live in completely different worlds (formal vs
-geometric) and neither implies the other. This is why the whole
-apparatus -- surface Quot schemes, Hilbert-scheme scalars,
-support induction -- is necessary. If we could prove non-membership
-from geometry alone, or vanishing from formalism alone, we
-wouldn't need either the detector or the surface machinery.
-
-That's the paper. The glossary follows.
-
-
-=====================================================
-GLOSSARY
-=====================================================
-
-The alphabetized terms appear throughout. Entries that are
-fairly technical (marked with *) are only needed if you want
-to read the formal proofs.
-
-A^* (X), Chow group of X
- Algebraic analog of cohomology. Cycles modulo rational
- equivalence, graded by codimension. Fact: A^*(X) is a
- graded commutative ring.
-
-A-curve
- Algebraic curve.
- A compact smooth complex curve of genus g. So it is
- simultaneously a Riemann surface of genus g and an
- algebraic curve over C.
-
-algebraic curve of genus g
- A compact smooth complex curve (Riemann surface) with
- g "holes".
-
-Aut Gamma (automorphism group of a graph Gamma)
- Permutations of half-edges preserving the graph structure
- and the marked vertices. Finite. Same for stable graphs.
-
-bar transform (*)
- The change of variables psi -> psi_bar and kappa -> kappa_bar
- that separates boundary divisors from the "root" part.
-
-bar_psi, bar_kappa (*)
- psi and kappa after subtracting all boundary contributions.
- The formal "root" cotangent and kappa.
-
-B_b (faithful Chow coefficient algebra) (*)
- A supercommutative rational algebra constructed in Part 5
- so that the tensor model of curve correspondences is
- faithful. Contains all Chow correspondences between curve
- powers, not just pairing diagrams.
-
-b (in the surface Quot section)
- The genus of the smooth curve family underlying the
- surface. The "b-1" appears in base codimension bounds
- because the surface Quot relation holds modulo that.
-
-base-support quotient (*)
- The quotient of A^* by classes vanishing off a closed
- subset of codimension at least a. Used to track loss
- of information on parameter families. Not numerical
- equivalence.
-
-beta_J^st, beta_{h,B} (bridge class)
- A codimension-two class summing over one-edge unmarked
- graphs of the relevant type, subdivided by a
- genus-zero vertex carrying the three labels.
-
-Chow group (see A^*)
- Algebraic cycles modulo rational equivalence.
-
-CohFT
- Cohomological Field Theory. An algebraic structure encoding
- tautological relations from Gromov-Witten theory. Not
- needed for our proof, but historically crucial for
- Pixton validity.
-
-compact type
- A boundary stratum with only compact-type stable curves
- (no separating nodes). M^ct_{g,n} is open inside Mbar_{g,n}.
-
-coproduct (*)
- The adjoint of multiplication with respect to the
- Frobenius pairing. Delta_L: V -> V^{(x)L} for a label set L.
-
-cycle class map cl
- A^* (X) -> H^{2 *}(X), the map from Chow groups to
- singular cohomology (over Q). Used for Corollary 2.2.
-
-D_T (tail divisor)
- A codimension-one boundary divisor on Mbar_{g,n}
- consisting of curves where the labels in T sit on a
- rational tail attached to the rest by a single node.
-
-D_bar_{ab}
- Sum over all T containing {a, b} of D_T.
-
-Delta_C, Delta_C^pure (*)
- Small diagonal in C x_S C x_S C, and its projection via
- eta_C in each of the three curve slots. Delta_C^pure is
- a pure odd-class in the tensor model.
-
-Delta_e (edge series) (*)
- Pixton's edge function in T, built from A(T), B(T) with
- the identity A(T)B(-T)+A(-T)B(T)+2=0.
-
-e'_C (*)
- e_C - (1/2) pi^*(a_C). A corrected degree-one divisor
- class. Ensures the "hyperbolic plane" decomposition works.
-
-eta (*)
- [Delta_C] - e'_{C,1} - e'_{C,2}. A projector onto the
- odd symplectic summand.
-
-eta_C (*)
- Same as eta but for a smooth core curve pi: C -> S.
-
-exterior algebra
- Lambda_Q(...) means: multiply symbols antisymmetrically.
- Square of any odd generator is zero.
-
-F^0_{h,B} (corrected cubic)
- The raw boundary cubic on the weighted compactification:
- [1=2=3] - (1/2) beta_{h,B} + (1/2) sum_p delta_{p,123} psi_p.
-
-F_J^st (three-slot factor)
- D_bar_{ab} * D_bar_{ac} - (1/2) beta_J^st, the building
- block of Y.
-
-Faber-Zagier
- An early system of relations among kappa classes on
- M_g (smooth curves). Its proving (Pandharipande-Pixton
- 2021) motivated Pixton's extension to Mbar.
-
-Frobenius algebra (*)
- An algebra V with a nondegenerate pairing (the integral),
- such that V is self-dual for that pairing. The odd
- symplectic examples here carry both a symplectic form
- and an alternating cubic form.
-
-g, n (stable pair parameters)
- g = genus, n = number of marked points. Stability is
- 2g-2+n > 0. The paper's counterexample uses g = 10^60
- and n = 3*C(g,3).
-
-Gorenstein property
- The tautological ring R^*(Mbar_{g,n}) is Gorenstein if
- multiplication R^d x R^{top-d} -> R^{top} is a perfect
- pairing. This is a "nice geometric property". It is
- independent from Pixton completeness.
-
-Gross-Schoen cycle (*)
- A corrected small diagonal in the triple product of a
- pointed curve. The prototype of the projected small
- diagonal in Part 5.
-
-Hassett (weighted) moduli space
- Mbar_{g,(w_1,...,w_n)} for positive rational weights
- summing to at most 1. Allows light weights to collide.
- Named after Brendan Hassett.
-
-heavy / light marks
- In weighted stable curves, "heavy" marks (weight 1) are
- kept separate; "light" marks (weight epsilon) may collide.
-
-Hilbert scheme of points (X^{[r]})
- Moduli of finite flat subschemes of length r. Smooth if
- X is a smooth surface. Used in the surface Quot relation.
-
-kappa class
- A class measuring how the marked points wrap around a
- curve. kappa_j = pi_*(psi^{j+1}) where pi forgets an
- extra mark. In the strata algebra: kappa_j is a formal
- symbol at each vertex.
-
-ker q_{g,n} (the kernel)
- The set of formal combinations in S_{g,n} mapping to zero
- under q_{g,n}. These are the actual relations.
-
-lambda
- (2g-2) e in the odd Frobenius algebra. Plays the role of
- the "handle element" at a curve. Properties: lambda^2 = 0.
-
-Losev-Manim space
- Mbar_{0,(1,1,epsilon,epsilon,epsilon)}. The weighted
- chain space with two heavy endpoints and three lights.
- In Lemma 6.3, it is the first blowup P^2 of the
- triple-exceptional locus.
-
-M_{g,n}, Mbar_{g,n}
- Moduli of smooth curves / stable curves of genus g with
- n marked points.
-
-marked point
- A distinguished labeled location on a curve.
-
-node
- A transverse self-intersection of a curve.
-
-Pixton relation R(g,n,r;sigma,a)
- The formal sum over stable graphs of [Gamma, P_Gamma] /
- |Aut Gamma| built from the recipe of Section 1.7.
-
-Pixton span P_{g,n}
- The formal span of all Pixton relations, multiplied and
- glued by arbitrary auxiliary strata. Already an ideal
- (Pixton Prop 1).
-
-psi class
- The cotangent class at a marked point (or at a
- half-edge). psi_i in A^1(Mbar_{g,n}).
-
-q_{g,n} (realization map)
- S_{g,n} -> A^*(Mbar_{g,n}), the formal-to-geometric map
- that sends [Gamma, gamma] to xi_{Gamma,*} gamma.
- Unnormalized (no Aut denominators).
-
-Quot scheme
- Moduli of quotients of a fixed coherent sheaf. In our
- surface setting, parametrizes rank-zero quotients of
- V = K + O_X with fixed determinant and ch_2.
-
-rational tails graph (*)
- A tree with one genus-g root and all other vertices
- rational, every rational branch having >= 2 descendant
- labels. Not the same as "curves with rational tails".
-
-rational tail divisor
- Same as a tail divisor D_T (see above).
-
-S_{g,n} (strata algebra)
- The free Q-vector space on decorated stable graphs modulo
- isomorphism, with per-vertex dimension truncation.
-
-small diagonal
- The diagonal Delta in C x C x C (three copies). Its
- "projection" via eta corresponds to a modified diagonal
- of Gross-Schoen type.
-
-stable graph
- A graph with genus-labelled vertices whose dual graph
- is the combinatorial shadow of a stable curve.
-
-Stable pair (g,n)
- 2g - 2 + n > 0. Necessary and sufficient for the moduli
- space Mbar_{g,n} to exist as a nonempty stack.
-
-support induction
- The method of reducing the codimension-a support of a
- cycle until it vanishes, used in Part 6.
-
-symplectic space (*)
- A vector space with a nondegenerate antisymmetric form
- < , >. In the odd setting, the form provides the pairing
- on V's middle summand.
-
-tautological (class / ring / space)
- Classes built from psi and kappa and gluing. The full
- subring R^* = q(S) of A^*(Mbar_{g,n}) is the
- tautological ring.
-
-Theta = product_{i<j<k} theta_{ijk}
- The top exterior monomial. Nonzero in R. This is the
- value that t_n(Y) reduces to, up to a nonzero scalar.
-
-theta_{ijk}
- Odd generator in the exterior ring R for each increasing
- triple of colors. Alternating in i,j,k.
-
-t_0 = 25 chi - 1 (chi = (b-1)^2)
- The power appearing in the surface bound: T_gamma^{t_0}=0.
-
-t_m (formal detector) (*)
- The recursive Q-linear map S_{g,m} -> V^{xm} that kills
- the Pixton span but satisfies t_n(Y) != 0.
-
-X_{h,B}(k) (weighted space)
- Hassett space with k light weights added to heavy data (h,B).
-
-Y (the counterexample class)
- Definition 3.4. Sum over signed permutations of the color
- word of product_J F_J^st. Lives in S_{g,n}^{2D_*}.
-
-xi_Gamma (gluing map)
- product_{v} Mbar_{g_v,n_v} -> Mbar_{g,n}. The map defining
- q_{g,n}.
-
-zeta_v (parity variable) (*)
- Zeta_v^2 = 1 at a vertex v. Controls the parity of kappa
- classes in Pixton's cycle formula.
-
-=====================================================
-END
-=====================================================
-
-
-The mathematics of the actual paper is much denser, but the
-ideas you just absorbed -- formal vs geometric, kernel vs
-generators, support induction, faithful tensor tests -- are
-exactly the tools mathematicians use at the research edge.
-The rest is filling in conventions.
-
-If you want to go further, useful starting points:
-
- - The moduli space of curves (Harris-Morrison,
- "Moduli of Curves"). For the geometric picture.
-
- - Intersection theory (Fulton, "Intersection Theory").
- For the algebraic machinery.
-
- - The Pandharipande-Pixton-Zvonkine papers
- (arXiv:1303.1043) for 3-spin relations.
-
- - Pixton's original (arXiv:1207.1918) for the
- relation recipe.
-
- - This paper's original LaTeX sources for the
- formal details.
-End of the rewrite.