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Welschinger--Witt invariants

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Pith's one-line read The paper proves that for k-rational del Pezzo surfaces of degree at least 6, quadratic Gromov–Witten invariants equal Welschinger–Witt invariants, and conjectures the equality for all rational surfaces.

desk verdict Welschinger–Witt invariants: a genuinely new bridge between real and quadratic Gromov–Witten counts, proved for degree ≥6 but conditional on three unpublished preprints. read the letter →

arxiv 2509.04172 v1 pith:NAAZN7Q4 submitted 2025-09-04 math.AG math.KTmath.SG

Erwan Brugallé , Johannes Rau , Kirsten Wickelgren This is my paper · ORCID
classification math.AGmath.KTmath.SG MSC 14N3514P9911E04
keywords WelschingerinvariantsquadraticGromov–WittenWittdelPezzosurfacesβ-integralityfloordiagramsrealenumerativegeometryA1-homotopytheory
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Welschinger invariants are signed counts of real rational curves through point constraints; quadratic Gromov–Witten invariants are their analogues over arbitrary fields of characteristic not 2 or 3, counting curves with values in the Witt ring of quadratic forms. This paper packages Welschinger invariants into β-integral Witt invariants (Welschinger–Witt invariants) and conjectures that every quadratic Gromov–Witten invariant of a k-rational del Pezzo surface is just the evaluation of one of these packages. The conjecture gives a uniform computation of all quadratic Gromov–Witten invariants of rational surfaces from the classical real numbers. The paper proves the conjecture for surfaces of degree at least 6, by showing the quadratic invariants are β-integral using the quadratic Abramovich–Bertram formula and floor diagram counts.

What carries the argument

Witt invariants of étale algebras in the sense of Serre: natural transformations Etn → W valued in Witt rings of quadratic forms. The paper uses the β-basis (βi) of the ring of such invariants; a Witt invariant is β-integral when its β-coefficients are integers. The Welschinger–Witt invariant V Wn,d is built from Welschinger invariants by the triangle recursion (a discrete version of Welschinger's formula), and the proof that quadratic Gromov–Witten invariants are also β-integral passes through essential marked floor diagrams whose quadratic multiplicities are expressed in the same β calculus.

What would settle it

Take k = F7, X = P^2_k, class d = 4 (so n0 = 11), and A = E_2 × E_3 × E_5 × E_7 × F7 in Et_11(F_7). Compute Q_{P^2,d,F7}(A) by enumerating the essential marked floor diagrams of Theorem 6.7 and sum their quadratic multiplicities over F7; compute V W_4(A) = 8β_1(A) + 2β_2(A) + β_3(A) with the β-basis of Section 2. The conjecture (and Theorem 6.1 for this case) holds exactly when the two Witt classes in W(F_7) agree.

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Extended reading notes

Core claim

Central claim (Theorem 6.1): for blow-ups of P^2 in at most three points, the quadratic Gromov–Witten invariant Q_{n,d} — a partially defined Witt invariant — is the restriction of the Welschinger–Witt invariant V W_{n,d}. This is a case of Conjecture 5.14: Q_{X,D,k}(A0) = V W_{X,D}(A0) for every k-rational del Pezzo surface. The authors construct V W_{n,d} as a β-integral Witt invariant whose multireal values are Welschinger invariants, and show quadratic Gromov–Witten invariants are unramified Witt invariants away from 2 and 3. The proof reduces to toric surfaces via the quadratic Abramovich–Bertram formula and concludes β-integrality by floor diagram counts.

Load-bearing premise

The proof relies on Theorem 6.7: that quadratic Gromov–Witten invariants of toric del Pezzo surfaces are computed by summing the quadratic multiplicities of essential marked floor diagrams, with the large-characteristic assumption of the underlying floor-diagram computation removed.

Editorial extensions

If this is right

  • If the conjecture is right, quadratic Gromov–Witten invariants of all k-rational del Pezzo surfaces are computed by the β-integral Welschinger–Witt invariants, whose inputs are only the real curve counts.
  • The equality implies deformation invariance of quadratic Gromov–Witten invariants for rational del Pezzo surfaces, since V W depends only on the étale algebras of the points.
  • Quadratic invariants in positive characteristic would be forced to agree with characteristic-zero values, a property the authors note is expected from construction.
  • The floor-diagram formula provides an effective algorithm: essential marked floor diagrams with Witt-valued multiplicities compute both sides explicitly.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If Conjecture 5.14 holds, every quadratic Gromov–Witten invariant of a k-rational surface is determined by signed counts of real curves, so computations over finite fields or p-adic fields can be replaced by real enumerative geometry.
  • The β-integrality of Welschinger–Witt invariants imposes divisibility and congruence conditions on Welschinger invariants themselves—essentially a Witt-valued refinement of Welschinger's formula—that could be tested numerically on existing tables.
  • The same triangle construction may apply to Welschinger invariants of P^3 and other Fano manifolds (as the paper sketches), predicting quadratic Gromov–Witten invariants of higher-dimensional varieties once the quadratic theory exists.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper proposes a conjectural relationship between Welschinger invariants, which are signed counts of real rational curves, and quadratic Gromov–Witten invariants, which are counts taking values in Witt–Grothendieck rings over general fields of characteristic not 2 or 3. The authors construct multivariable unramified Witt invariants from Welschinger invariants (Theorem 4.3), prove that quadratic Gromov–Witten invariants are Witt invariants and are unramified under appropriate hypotheses (Theorems 5.4 and 5.7), and formulate Conjecture 5.14 asserting equality of these two types of invariants for rational del Pezzo surfaces. The paper proves this conjecture for del Pezzo surfaces of degree at least 6 (Theorem 6.1), equivalent to blowing up at most three points, by reducing first to toric surfaces via a quadratic Abramovich–Bertram formula and then using a floor-diagram computation. The central theorem, Theorem 6.1, is thus built on the partially proven Theorem 6.7, which removes the large-characteristic assumption from the floor-diagram identity of [JPMPR25].

Significance. If the main theorem and conjecture hold, the paper provides a striking structural link: all quadratic Gromov–Witten invariants of k-rational surfaces would be determined by Welschinger invariants, packaged as β-integral Witt invariants. The construction of multivariable Witt invariants from Welschinger invariants, the multireal-triangle calculus, and the explicit tables of β- and λ-coefficients are valuable contributions in their own right. The paper is ambitious and introduces new tools that are likely to be influential. However, the central claim is conditional on the unproven details of Theorem 6.7 and on the availability and correctness of several unpublished preprints. The paper is not fully self-contained, and the proof of the key floor-diagram identity is only sketched.

major comments (2)
  1. [Section 6.1, Theorem 6.7] Theorem 6.7 is load-bearing: Corollary 6.8 and therefore Theorem 6.1 rest on the claim that the floor-diagram sum computes the quadratic Gromov–Witten invariant Q'^{(m-s)}_d. The proof given is a single paragraph asserting that both sides are unramified away from S={2,3} and then invoking the characteristic-0 result [JPMPR25, Theorem 10.13]. This is not sufficient as written. To make the spreading-out argument work, the authors must verify: (1) the left-hand side is unramified, which requires checking that the toric surface X_{k^3} has a smooth proper model over Z[1/6] and that Hypothesis 5.1 holds for the relevant divisor class; (2) the right-hand side is an S-integral Witt invariant on Sq_s — this is plausible since μ(D,φ) ∈ Z[t_1,…,t_s], but it is not demonstrated; and (3) the cited [JPMPR25, Theorem 10.13] indeed proves the identity as an equality of Witt invariants over Q, i.e., for
  2. [Section 6.2, Lemma 6.14] The positive-characteristic reduction in the proof of Lemma 6.14 is too compressed. The sentence 'Since the lemma holds in characteristic 0, it follows that the lemma holds for k perfect of positive characteristic as well' is only valid if both sides of the asserted equality are known to be unramified over the chosen DVR R, so that equality over the fraction field descends to the residue field. The proof should explicitly invoke Proposition 5.6, identify the model over R (the blow-up Bl_{E_{\tilde δ}×R} P^2_R), and check that the hypotheses of Proposition 5.6 are satisfied. As written, the descent step is asserted rather than demonstrated, which is a gap in a key reduction used in the proof of Theorem 6.1.
minor comments (5)
  1. [Throughout] There are several typos and inconsistencies: 'Weslchinger' for 'Welschinger' in Remark 4.12, 'Gomov' for 'Gromov' in Conjecture 5.14, and a garbled name in the acknowledgments ('BenoîtV WBertrand'). The notation Et_n(K) / Etn(K) is used inconsistently; please unify.
  2. [Section 2.3, Definition 2.15] The diagram in Definition 2.15 is hard to read: the placement of the isomorphism signs (∼=) is ambiguous. It should be clarified which arrows are isomorphisms and which are merely functoriality maps.
  3. [Section 6.2, proof of Theorem 6.1] After proving β-integrality of Q_{n,d,Q}, the proof states that the invariant 'must equal V W_{n,d}' by Theorem 4.3. This also uses Lemma 5.13 to identify the multireal values of Q_{n,d,Q} with those of V W_{n,d}; this step should be stated explicitly.
  4. [Section 6.1, proof of Corollary 6.8] The proof uses the identities t_j^2 = 2t_j and the identification of the basis {t_J} with the β-basis. These facts are used without reference; a pointer to Theorem 2.5 or Lemma 2.14 would help the reader.
  5. [General] The paper relies heavily on the unpublished preprints [KLSW23a], [BW25], and [JPMPR25]. While this is not a mathematical error, the authors should state in the introduction or in a remark exactly which results from these preprints are used and whether they are available in final form. This is particularly important for Theorem 6.7, which is a nontrivial generalization of [JPMPR25, Theorem 10.13].

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the equality of quadratic Gromov–Witten and Welschinger–Witt invariants is proved via independent floor-diagram and Abramovich–Bertram inputs, not by construction or self-citation alone.

full rationale

The paper's central claim, Theorem 6.1, is that a partially defined quadratic Gromov–Witten invariant Q_{n,d} equals the Welschinger–Witt invariant V W_{n,d} for n1+...+nr ≤ 3. Neither side is defined in terms of the other: V W_{n,d} is built from Welschinger invariants via β-integral Witt invariants (Theorem 4.3), while Q_{X,D} is defined via the A1-degree of an oriented evaluation map (Section 5.1, following [KLSW23a]) and is shown independently to be a Witt invariant and unramified away from 2 and 3 (Theorems 5.4 and 5.7). The equality over R is an external theorem (Welschinger invariants as signatures of quadratic Gromov–Witten invariants, Lemma 5.13), and the hard part is β-integrality of Q, which is established by reducing to toric surfaces via the quadratic Abramovich–Bertram formula ([BW25], Lemma 6.14) and then applying the floor-diagram computation of [JPMPR25] (Theorem 6.7). The removal of the 'sufficiently large characteristic' assumption in Theorem 6.7 is a valid unramified specialization argument: both sides are unramified away from {2,3}, so equality in characteristic 0 implies equality in every residue characteristic not 2 or 3. This is not a fit or a renaming, nor does it assume the target equality. The self-citations to [KLSW23a], [KLSW23b], and [BW25] are used as external mathematical inputs; the cited statements do not include Conjecture 5.14 or Theorem 6.1, and the paper reproduces or sketches the relevant reductions rather than merely citing its own conclusion. Genuine limitations—such as the restriction to perfect fields, the lack of full deformation invariance, and the exclusion of characteristics 2 and 3—are explicitly acknowledged but are correctness or scope concerns, not circularity. No step was found in which the claimed prediction reduces by definition to the input data.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces new invariants, but these are explicitly constructed from existing mathematical objects rather than postulated entities. No parameters are fitted: the beta-coefficients are integers determined by classical Welschinger invariants, which are inputs from prior literature. The axioms listed are the main external facts on which the construction and proof rest.

assumptions (5)
  • standard math Witt invariant structure theorem: Inv_k(n) is a free W(k)-module with beta-basis (Theorem 2.5, from [GMS03, Section 29]).
    Used throughout to define beta-coefficients and beta-integrality.
  • domain assumption Existence and base-change properties of quadratic Gromov-Witten invariants, including the A1-degree of twisted evaluation maps over non-perfect fields (Definition 5.3, Lemma 5.2).
    Provided by [KLSW23a,b] (preprint, one author overlaps); the paper extends these to non-perfect fields.
  • domain assumption The quadratic Abramovich-Bertram formula of [BW25] applies to the blow-up families in Lemma 6.14.
    Used to reduce the proof of the main theorem to toric surfaces; [BW25] is a preprint by two of the present authors.
  • domain assumption The floor diagram computation of [JPMPR25, Theorem 10.13] computes Q'_{n,d}, with the large-characteristic assumption removed (Theorem 6.7).
    External preprint supplies the tropical multiplicity count used to prove beta-integrality in Corollary 6.8.
  • domain assumption Welschinger invariants satisfy the real Abramovich-Bertram formula (Theorem 4.1, from [Bru20, Proposition 2.3]).
    Used to assemble Welschinger invariants into beta-integral multireal values.

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Pith. "Pith review of Welschinger--Witt invariants." pith.science (2026). https://pith.science/paper/NAAZN7Q4

@misc{pith2026250904172,
  author       = {Pith},
  title        = {Pith review of: Welschinger--Witt invariants},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NAAZN7Q4}},
  note         = {Machine review of arXiv:2509.04172}
}
abstract

Welschinger invariants are signed counts of real rational curves satisfying contraints. Quadratic Gromov--Witten invariants give such counts over general fields of characteristic different from 2 and 3. For rational del Pezzo surfaces over a field, we propose a conjectural relationship between Welschinger and quadratic Gromov--Witten invariants. We construct multivariable unramified Witt invariants, in the sense of Serre, from Welschinger invariants and call them Welschinger--Witt invariants. We show that quadratic Gromov--Witten invariants are also Witt invariants and control their ramification. We then conjecture an equality between these Witt invariants, in particular giving a conjectural computation of all the quadratic Gromov--Witten invariants of $k$-rational surfaces. We prove this conjecture for $k$-rational del Pezzo surfaces of degree at least 6.

Figures

Figures reproduced from arXiv: 2509.04172 by the authors.

Figure 6.1
Figure 6.1. X = P 2 3,0 of P 2 R in 3 real points and the divisor D = d0L − d1E1 − d2E2 − d3E3. Given a floor diagram D = (G, θ), by abuse of notation we also denote by D the disjoint union of Ve(G), Ed(G), Ed+∞(G) and Ed−∞(G). This is a partially ordered set of size n = (d0 − d1) + (d0 − d1 − 1) + d1 + (d0 − d2 − d3) = 3d0 − d1 − d2 − d3 − 1, where x ⩽ y if there is an oriented path in G connecting x to y. Definition 6.4. Fix … view at source ↗

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