Refined curve counting on complex surfaces

Lothar G\"ottsche; Vivek Shende

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Refined curve counting on complex surfaces

Lothar Göttsche, Vivek Shende

Algebraic Geometry · Submitted 9 Aug 2012

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The authors' main claim We define refined invariants which "count" nodal curves in sufficiently ample linear systems on surfaces, conjecture that their generating function is multiplicative, and conjecture explicit formulas in the case of K3 and abelian surfaces.

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What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.47 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.07 · compares with prior results 0.89 · mentions code or data availability 0.02.

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We define refined invariants which "count" nodal curves in sufficiently ample linear systems on surfaces, conjecture that their generating function is multiplicative, and conjecture explicit formulas in the case of K3 and abelian surfaces. We also give a refinement of the Caporaso-Harris recursion, and conjecture that it produces the same invariants in the sufficiently ample setting. The refined recursion specializes at y = -1 to the Itenberg-Kharlamov-Shustin recursion for Welschinger invariants. We find similar interactions between refined invariants of individual curves and real invariants of their versal families.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We define refined invariants which "count" nodal curves in sufficiently ample linear systems on surfaces, conjecture that their generating function is multiplicative, and conjecture explicit formulas in the case of K3 and abelian surfaces.	contribution 1.00	0.92	0.50	speculative 0.82
s1	We also give a refinement of the Caporaso-Harris recursion, and conjecture that it produces the same invariants in the sufficiently ample setting.	contribution 0.96	0.89	0.12	speculative 0.90
s2	The refined recursion specializes at y = -1 to the Itenberg-Kharlamov-Shustin recursion for Welschinger invariants.	result 0.76	0.57	0.10	asserted 1.00
s3	We find similar interactions between refined invariants of individual curves and real invariants of their versal families.	result 0.83	0.90	0.07	asserted 1.00

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Original abstract

We define refined invariants which "count" nodal curves in sufficiently ample linear systems on surfaces, conjecture that their generating function is multiplicative, and conjecture explicit formulas in the case of K3 and abelian surfaces. We also give a refinement of the Caporaso-Harris recursion, and conjecture that it produces the same invariants in the sufficiently ample setting. The refined recursion specializes at y = -1 to the Itenberg-Kharlamov-Shustin recursion for Welschinger invariants. We find similar interactions between refined invariants of individual curves and real invariants of their versal families.

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References

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8 arXiv papers cite this in 53 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 39 sentences without a confident label

Builds on it 1 paper

This note grew out of an attempt to reprove their results in terms of tropical geometry, and use this approach to extend them to the refined Severi degrees defined in [ref] , [ref] .

Fock spaces and refined Severi degrees · Introduction · 0.82

Uses its method or tool 2 papers

In [ref] a refined Caporaso-Harris type recursion formula is used to define Laurent polynomials $N^(S,L),δ(y)$, which the authors call refined Severi degrees.

Refined curve counting with tropical geometry · Refined Severi degrees · 0.81
As explained in [ref] , for a linear subsystem $^δ⊂|L|$ such that the relative Hilbert schemes $^[n]_[S,L],δ →^δ$ are all smooth – e.g., a general $δ$-dimensional linear subsystem when $L$ is $δ$-very-ample [ref] – we may extract the $χ_-y$ genera by taking a residue:

The chi-y genera of relative Hilbert schemes for linear systems on Abelian and K3 surfaces · Universality arguments · 0.89

Uses its data or values 1 paper

In [ref] we have computed $N^(S,L),δ(y)$ for all $(S,L)$ and all $δ≤10$.

Refined curve counting with tropical geometry · Refined node polynomials · 0.81

Compares with it 2 papers

This result was also obtained in [ref] and [ref] .

Refined curve counting with tropical geometry · Coefficient polynomiality of refined Severi degrees · 0.86
Then we define[ Note this differs from [ref] by the normalization by $y^-/2$. ] [equation]

The chi-y genera of relative Hilbert schemes for linear systems on Abelian and K3 surfaces · Universality arguments · 0.88

Cites it as background 5 papers

In this section we review the definition of the closely related notions of the refined invariants and the refined Severi degrees from [ref] .

Refined curve counting with tropical geometry · Refined invariants and refined Severi Degrees · 1.00
In [ref] , [ref] irreducible refined Severi degrees $N_0^(S,L),δ(y)$, $N^Δ,δ_0(y)$ are introduced and studied.

Fock spaces and refined Severi degrees · Main theorems · 0.98
Recently Block and Göttsche [ref] (see also the accounts in [ref] section 6 and [ref] section 1) introduced a refined tropical count for plane tropical curves, where the usual Mikhalkin multiplicity is replaced by a function taking values in Laurent polynomials in one variable.

Block-G\"ottsche invariants from wall-crossing · Introduction · 0.99

What it cites · 47 sentences without a confident label

Builds on 3 papers

The following result lifts [ref] from Euler characteristics to $$; the proof is essentially the same.

Stable pairs and BPS invariants · Refined invariants · 0.99
Using the tropical interpretation of the $N^L,δ(α,β)$ of [ref] using refined multiplicities (see also Section [ref]), this result has been generalized in [ref] to arbitrary toric surfaces.

On Block-Goettsche multiplicities for planar tropical curves · The refined invariants at $y=0$ · 0.85
The first equality was asserted in [ref] in the case of a locally versal family, but in fact the same argument applies in the above generality.

Hilbert schemes of points on a locally planar curve and the Severi strata of its versal deformation · Refined invariants of linear systems · 0.89

Uses the method or tool of 1 paper

Using the refined multiplicity, in [ref] the $N^L,δ_(Σ)$ are studied using methods similar to those employed in [ref] for the nonrefined Severi degrees.

Computing Node Polynomials for Plane Curves · Refined, real, and tropical · 0.89

Compares with 1 paper

The statement of Theorem [ref] in [ref] differs slightly; there are some signs owing to the use of $Ω$ rather than $T$, and it is formulated in terms of the space of “stable pairs” rather than the relative Hilbert scheme.

Curves on K3 surfaces and modular forms · The term of the deepest stratum · 1.00

Cites as support 3 papers

Then according to [ref] , the first $^[≤i]_D^i_0$ are all smooth, and if $i ≥δ$ then all $^[n]_D^i_0$ are smooth.

Hilbert schemes of points on a locally planar curve and the Severi strata of its versal deformation · Real invariants · 0.98
Then according to [ref] , the first $^[≤i]_D^i_0$ are all smooth, and if $i ≥δ$ then all $^[n]_D^i_0$ are smooth.

Euler number of the compactified Jacobian and multiplicity of rational curves · Real invariants · 0.98
The equality at $y = 1$ follows from [ref] .

On the G\"ottsche Threshold · Introduction · 0.96

Cites as background 6 papers

Thus far we have been discussing curves with a number of nodes small compared to the ampleness of the line bundle $L$; this is the regime to which the conjectures of [ref] and the arguments of [ref] apply.

Testing Ultrafast Two-Photon Spectral Amplitudes via Optical Fibres · Introduction · 0.84
Mathematically, the refined theory is supposed to correspond [ref] to the motivic DT theory [ref] ; the lack of a worldsheet definition corresponds to the fact that we do not know how to correspondingly refine the Gromov-Witten invariants.

Stability structures, motivic Donaldson-Thomas invariants and cluster transformations · Introduction · 0.84
Mathematically, the refined theory is supposed to correspond [ref] to the motivic DT theory [ref] ; the lack of a worldsheet definition corresponds to the fact that we do not know how to correspondingly refine the Gromov-Witten invariants.

Refined, Motivic, and Quantum · Introduction · 0.84

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