Singular Welschinger invariants

Eugenii Shustin

arxiv: 1907.00461 · v1 · pith:24DJANYVnew · submitted 2019-06-30 · 🧮 math.AG

Singular Welschinger invariants

Eugenii Shustin This is my paper

Pith reviewed 2026-05-25 12:31 UTC · model grok-4.3

classification 🧮 math.AG

keywords singular Welschinger invariantsreal plane curve singularitiesnodal deformationsnodal-cuspidal deformationsWelschinger signsenumerative geometryreal algebraic curves

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The pith

Welschinger signs produce deformation-invariant counts for real nodal deformations of plane curve singularities.

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

The paper proposes an invariant enumeration of nodal and nodal-cuspidal real deformations of singularities on real plane curves. It achieves this by attaching Welschinger signs to each deformation in the count. The resulting numbers are presented as a local counterpart to the Welschinger invariants that count real rational plane curves globally. A reader would care because the construction supplies counts that remain stable when the underlying singularity undergoes real deformation.

Core claim

We suggest an invariant way to enumerate nodal and nodal-cuspidal real deformations of real plane curve singularities. The key idea is to assign Welschinger signs to the counted deformations. Our invariants can be viewed as a local version of Welschinger invariants enumerating real plane rational curves.

What carries the argument

Assignment of Welschinger signs to nodal and nodal-cuspidal real deformations of the singularities, making the signed count invariant.

If this is right

The signed enumeration stays constant for any real deformation of a given singularity.
The construction applies equally to both nodal and nodal-cuspidal cases.
The local invariants mirror the global Welschinger counts for rational curves.
The method yields well-defined numbers attached directly to the singularity type.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

These local invariants could supply additive contributions when computing global Welschinger numbers for curves with prescribed singularities.
Explicit formulas depending only on the analytic type of the singularity may exist and could be computed without reference to an ambient curve.
The same sign-assignment technique might extend to other classes of real curve singularities or to higher-dimensional varieties.

Load-bearing premise

Welschinger signs can be assigned to the deformations such that the resulting enumeration is invariant under real deformations of the singularities.

What would settle it

A concrete real plane curve singularity together with a real deformation of it for which the signed count of nodal deformations changes.

read the original abstract

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Shustin proposes local Welschinger invariants by signing nodal and nodal-cuspidal deformations of real plane curve singularities, but the abstract supplies no definitions or checks for invariance.

read the letter

The central point is a suggested construction of singular Welschinger invariants that count real deformations of plane curve singularities with signs drawn from the global Welschinger theory, presented as a local analog for rational curves. The new element is the explicit move to localize the signed count to singularities rather than global curves. This framing connects directly to existing work on real enumerative invariants and could be useful for people already working in that corner of real algebraic geometry. The paper states the objects to be counted (nodal and nodal-cuspidal deformations) and the signing strategy without adding extra parameters or invented entities. That keeps the proposal clean on its own terms. The stress-test note finds no visible internal contradiction in the outline, which is fair given what is shown. The main limitation is that the abstract gives no definition of the signs, no argument that the resulting number stays constant under real deformation of the singularity, and no sample calculation. Without those steps the invariance claim cannot be checked, so the soundness score from the abstract alone is low. If the full text supplies a consistent sign rule and a deformation argument, the gap closes; if not, the construction stays formal. The work is written for specialists already comfortable with Welschinger invariants and real plane curves. A reader outside that subfield will not get much from it. It shows straightforward engagement with the literature rather than incoherence. I would bring it to a reading group only if the group already covers real enumerative geometry. I would not cite it until the details are verified. It is worth sending to referees so they can examine whether the sign assignment actually produces an invariant.

Referee Report

1 major / 0 minor

Summary. The paper proposes an invariant enumeration of nodal and nodal-cuspidal real deformations of real plane curve singularities. The construction assigns Welschinger signs to the counted deformations and is presented as a local analogue of the Welschinger invariants that enumerate real plane rational curves.

Significance. If the signed counts are shown to be invariant, the construction would supply a local enumerative tool in real algebraic geometry that parallels the global Welschinger theory and could be used to study the behavior of real curves near singularities.

major comments (1)

The manuscript states the suggestion of assigning Welschinger signs to obtain an invariant count but supplies neither an explicit definition of those signs for nodal/nodal-cuspidal deformations nor a verification that the resulting enumeration is deformation-invariant. This is the load-bearing step for the central claim.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report and for identifying the central requirement for establishing the proposed invariants. We address the major comment below.

read point-by-point responses

Referee: The manuscript states the suggestion of assigning Welschinger signs to obtain an invariant count but supplies neither an explicit definition of those signs for nodal/nodal-cuspidal deformations nor a verification that the resulting enumeration is deformation-invariant. This is the load-bearing step for the central claim.

Authors: We agree that the submitted manuscript proposes the idea of assigning Welschinger signs to nodal and nodal-cuspidal real deformations but does not supply an explicit definition of the signs or a verification of deformation invariance. This is a genuine gap in the current version. In the revised manuscript we will add a precise definition of the signs (based on the local real topology of the deformations) together with a proof that the signed count is invariant under deformations of the singularity. This revision will make the local analogue to global Welschinger invariants fully rigorous. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper defines an enumeration of real deformations of plane curve singularities by assigning Welschinger signs, claiming the resulting count is invariant under real deformations and forms a local version of known Welschinger invariants. No load-bearing step reduces by construction to a fitted parameter, self-definition, or self-citation chain; the sign assignment and invariance claim constitute an independent construction rather than a renaming or tautological reduction of inputs. The derivation is self-contained against external benchmarks (existing Welschinger theory) with no evident circular patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no information on free parameters, axioms, or invented entities.

pith-pipeline@v0.9.0 · 5547 in / 987 out tokens · 50511 ms · 2026-05-25T12:31:11.594517+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages

[1]

N. A’Campo. Le groupe de monodromie des singularit´ es is ol´ ees des courbes planes, I. Math. Ann. 213 (1975), 1–32

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M. Artin. Versal deformations and algebraic stacks. Invent. Math. 27 (1974), 165–189

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Beauville

A. Beauville. Counting rational curves on K3 surfaces. Duke Math. J. 97 (1999), no. 1, 99–108

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Casas-Alvero

E. Casas-Alvero. Singularities of Plane Curves . Cambridge Univ. Press, Cambridge, 2000

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Chiang-Hsieh and J

H.-J. Chiang-Hsieh and J. Lipman. A numerical criterion for simulta- neous normalization. Duke Math. J. 133 (2006), no. 2, 347–390

work page 2006
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S. Diaz. Irreducibility of the equiclassical locus. J. Diﬀ. Geom. 29 (1989), 489–498

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Diaz and J

S. Diaz and J. Harris. Ideals associated to deformations of singular plane curves. Trans. Amer. Math. Soc. 309 (1988), no. 2, 433–468

work page 1988
[8]

Fantechi, L

B. Fantechi, L. G¨ ottsche, and D. van Straten. Euler numb er of the compactiﬁed Jacobian and multiplicity of rational curves. J. Algebraic Geom. 8 (1999), 115–133

work page 1999
[9]

Greuel G.-M., C

G.-M. Greuel G.-M., C. Lossen C., and E. Shustin. Introduction to singularities and deformations . Springer, NY, 2007

work page 2007
[10]

Greuel G.-M., C

G.-M. Greuel G.-M., C. Lossen C., and E. Shustin. Singular algebraic curves. Springer, NY, 2018

work page 2018
[11]

Itenberg, V

I. Itenberg, V. Kharlamov, and E. Shustin. Welschinger invariants re- visited. Analysis Meets Geometry: A Tribute to Mikael Passare. Trends in Math., Birkh¨ auser, 2017, pp. 239–260

work page 2017
[12]

Leviant and E

P. Leviant and E. Shustin. Morsiﬁcations of real plane c urve singulari- ties. J. Singularities 18 (2018), 307–328

work page 2018
[13]

D. Mumford. Algebraic geometry I: complex projective varieties . (2nd ed.), Springer, 1976

work page 1976
[14]

Shevchishin

V. Shevchishin. On the local Severi problem. Int. Math. Res. Not. 2004, no. 5, 211–237. 18

work page 2004
[15]

V. Shende. The Hilbert schemes of points on a locally pla nar curve and the Severi strata of its versal deformation. Compositio Math. 148 (2012), no. 2, 531–547

work page 2012
[16]

Teissier

B. Teissier. The hunting of invariants in the geometry o f discriminants. Real and Complex Singularities, Oslo 1976 / P. Holm (ed.), Sijthoﬀ- Noordhoﬀ Publ., Alphen aan den Rijn, 1977, pp. 565–677

work page 1976
[17]

Welschinger

J.-Y. Welschinger. Invariants of real symplectic 4-ma nifolds out of re- ducible and cuspidal curves. Bull. Soc. math. France 134 (2006), no. 2, 287–325. 19

work page 2006

[1] [1]

N. A’Campo. Le groupe de monodromie des singularit´ es is ol´ ees des courbes planes, I. Math. Ann. 213 (1975), 1–32

work page 1975

[2] [2]

M. Artin. Versal deformations and algebraic stacks. Invent. Math. 27 (1974), 165–189

work page 1974

[3] [3]

Beauville

A. Beauville. Counting rational curves on K3 surfaces. Duke Math. J. 97 (1999), no. 1, 99–108

work page 1999

[4] [4]

Casas-Alvero

E. Casas-Alvero. Singularities of Plane Curves . Cambridge Univ. Press, Cambridge, 2000

work page 2000

[5] [5]

Chiang-Hsieh and J

H.-J. Chiang-Hsieh and J. Lipman. A numerical criterion for simulta- neous normalization. Duke Math. J. 133 (2006), no. 2, 347–390

work page 2006

[6] [6]

S. Diaz. Irreducibility of the equiclassical locus. J. Diﬀ. Geom. 29 (1989), 489–498

work page 1989

[7] [7]

Diaz and J

S. Diaz and J. Harris. Ideals associated to deformations of singular plane curves. Trans. Amer. Math. Soc. 309 (1988), no. 2, 433–468

work page 1988

[8] [8]

Fantechi, L

B. Fantechi, L. G¨ ottsche, and D. van Straten. Euler numb er of the compactiﬁed Jacobian and multiplicity of rational curves. J. Algebraic Geom. 8 (1999), 115–133

work page 1999

[9] [9]

Greuel G.-M., C

G.-M. Greuel G.-M., C. Lossen C., and E. Shustin. Introduction to singularities and deformations . Springer, NY, 2007

work page 2007

[10] [10]

Greuel G.-M., C

G.-M. Greuel G.-M., C. Lossen C., and E. Shustin. Singular algebraic curves. Springer, NY, 2018

work page 2018

[11] [11]

Itenberg, V

I. Itenberg, V. Kharlamov, and E. Shustin. Welschinger invariants re- visited. Analysis Meets Geometry: A Tribute to Mikael Passare. Trends in Math., Birkh¨ auser, 2017, pp. 239–260

work page 2017

[12] [12]

Leviant and E

P. Leviant and E. Shustin. Morsiﬁcations of real plane c urve singulari- ties. J. Singularities 18 (2018), 307–328

work page 2018

[13] [13]

D. Mumford. Algebraic geometry I: complex projective varieties . (2nd ed.), Springer, 1976

work page 1976

[14] [14]

Shevchishin

V. Shevchishin. On the local Severi problem. Int. Math. Res. Not. 2004, no. 5, 211–237. 18

work page 2004

[15] [15]

V. Shende. The Hilbert schemes of points on a locally pla nar curve and the Severi strata of its versal deformation. Compositio Math. 148 (2012), no. 2, 531–547

work page 2012

[16] [16]

Teissier

B. Teissier. The hunting of invariants in the geometry o f discriminants. Real and Complex Singularities, Oslo 1976 / P. Holm (ed.), Sijthoﬀ- Noordhoﬀ Publ., Alphen aan den Rijn, 1977, pp. 565–677

work page 1976

[17] [17]

Welschinger

J.-Y. Welschinger. Invariants of real symplectic 4-ma nifolds out of re- ducible and cuspidal curves. Bull. Soc. math. France 134 (2006), no. 2, 287–325. 19

work page 2006