[Paper Review] A tropical calculation of the Welschinger invariants of real toric Del Pezzo surfaces

Eugeniĭ Shustin|ArXiv.org|Jun 6, 2004

Polynomial and algebraic computation13 references5 citations

TL;DR

This paper establishes a tropical formula to compute Welschinger invariants for real toric Del Pezzo surfaces by relating them to the total multiplicity of tropical curves and lattice paths in the associated Newton polygon. The method provides a combinatorial, recursive algorithm based on consistent subdivisions of lattice polygons, yielding exact counts of real rational curves through conjugation-invariant point configurations, with the key result being a closed-form expression for Welschinger invariants via tropical geometry and lattice path enumeration.

ABSTRACT

The Welschinger invariants of real rational algebraic surfaces are natural analogues of the genus zero Gromov-Witten invariants. We establish a tropical formula to calculate the Welschinger invariants of real toric Del Pezzo surfaces for any conjugation-invariant configuration of points. The formula expresses the Welschinger invariants via the total multiplicity of certain tropical curves (non-Archimedean amoebas) passing through generic configurations of points, and then via the total multiplicity of some lattice path in the convex lattice polygon associated with a given surface. We also present the results of computation of Welschinger invariants, obtained jointly with I. Itenberg and V. Kharlamov.

Motivation & Objective

• To develop a combinatorial method for computing Welschinger invariants of real toric Del Pezzo surfaces, which are real analogues of genus-zero Gromov-Witten invariants.
• To address the lack of closed-form or recursive formulas for Welschinger invariants in the presence of both real and conjugate imaginary point configurations.
• To extend tropical enumerative geometry to handle conjugation-invariant configurations, thereby enabling the calculation of invariants that provide non-trivial lower bounds on real rational curve counts.
• To establish a precise correspondence between real rational tropical curves and algebraic curves via valuation and degeneration techniques, ensuring invariance under generic configurations.

Proposed method

• The method uses non-Archimedean valuation to lift complex algebraic curves over Puiseux series to tropical curves in R², preserving their enumerative properties.
• It constructs real rational tropical curves as limits of algebraic curves through generic configurations of real and conjugate imaginary points, using the valuation of coordinates.
• A recursive algorithm builds consistent subdivisions of the Newton polygon ∆ by adding triangles or parallelograms, ensuring the resulting tropical curve satisfies the equilibrium condition and has the correct type (r′, r′′, s′′).
• The Welschinger invariant is computed as the sum of weighted contributions from all consistent tropical curves, where weights depend on the number of solitary nodes in the corresponding algebraic curves.
• The algorithm incorporates constraints on vertex sets V and W to ensure only relevant tropical curves contribute, with non-zero contributions only when |σj| is odd for j ∈ V.
• The final formula expresses the Welschinger invariant as a sum over all sequences of lattice points in ∆ ∩ ℤ² ordered by a generic linear function λ, with consistent subdivisions yielding the total invariant.

Experimental results

Research questions

• RQ1 How can Welschinger invariants be computed for real toric Del Pezzo surfaces when the point configuration includes both real and conjugate imaginary points?
• RQ2 What is the tropical counterpart of the Welschinger invariant, and how does it relate to the enumeration of real rational curves?
• RQ3 Can a recursive, combinatorial algorithm based on lattice polygon subdivisions yield exact values of Welschinger invariants for such surfaces?
• RQ4 What conditions ensure that a tropical curve corresponds to a real algebraic curve with a specific number of solitary nodes, and how does this affect the invariant’s sign?
• RQ5 How does the tropical approach extend previous results valid only for all-real point configurations to the general case with conjugate pairs?

Key findings

• The Welschinger invariant W_r′′(Σ, D) is computed as a sum over consistent subdivisions of the Newton polygon ∆, with each subdivision contributing a weighted count based on tropical curve multiplicity.
• For real plane cubics (Δ = Conv{(0,0),(0,3),(3,0)}), the formula yields W_r′′(ℙ², 3L) = 8 − 2r′′, confirming known results via tropical enumeration.
• The method correctly accounts for the sign of the Welschinger invariant by tracking the parity of solitary nodes in the corresponding algebraic curves via tropical geometry.
• The invariant is independent of the choice of generic configuration in Ω_r′′(Σ, D), as proven via tropical degeneration and invariance under deformation.
• The construction shows that W_4(ℙ², 3L) = 0, and this is confirmed by the existence of a pencil of non-singular real cubics through four pairs of imaginary conjugate points.
• The algorithm successfully handles the case of conjugate imaginary points by incorporating constraints on vertex sets and ensuring that only subdivisions with odd-length edges contribute non-trivially.