[Paper Review] On refined count of rational tropical curves

Eugeniĭ Shustin|arXiv (Cornell University)|Dec 19, 2018

Polynomial and algebraic computation14 references4 citations

TL;DR

This paper introduces two new refined invariants for counting rational tropical curves: one for plane curves with an unmarked four-valent vertex, motivated by unicuspidal curve enumeration, and another for rational curves in Euclidean space of arbitrary dimension with specific constraints. The invariants are constructed by refining vertex weights using a formal parameter, and their invariance is proven via combinatorial identities in the Grassmannian of 2-planes, ensuring consistency under small deformations of constraints.

ABSTRACT

We address the problem of existence of refined (i.e., depending on a formal parameter) tropical enumerative invariants, and we present two new examples of a refined count of rational marked tropical curves. One of the new invariants counts plane rational tropical curves with an unmarked vertex of arbitrary valency. It was motivated by the tropical enumeration of plane cuspidal tropical curves given by Y. Ganor and the author, which naturally led to consideration of plane tropical curves with an unmarked four-valent vertex. Another refined invariant counts rational tropical curves of a given degree in the Euclidean space of arbitrary dimension matching specific constraints, which make the spacial refined invariant similar to known planar invariants.

Motivation & Objective

• To establish new refined tropical enumerative invariants that generalize existing planar invariants to curves with unmarked higher-valence vertices.
• To provide a refined count of rational tropical curves in R^m (m ≥ 2) under specific geometric constraints, extending planar invariants to higher dimensions.
• To resolve the challenge of refining numerical tropical invariants when curve weights depend on global features, by identifying settings where weights factorize into local components.
• To connect tropical invariants to algebraic-geometric counterparts, particularly in the context of unicuspidal curve enumeration.

Proposed method

• Define marked rational tropical curves in R^m as metric graphs with a map to R^m satisfying the balancing condition at vertices and marked points.
• Introduce a refined invariant by replacing standard vertex weights with their quantum analogues involving a formal parameter z.
• Prove invariance of the refined count under small perturbations of constraints by analyzing the sign of oriented 2-planes in the Grassmannian Gr(2, m).
• Use combinatorial identities involving wedge products of directing vectors (e.g., z^{a∧b} − z^{b∧a}) to verify invariance across different configurations of edges.
• Restrict to configurations where the weight of each curve splits into independent factors per vertex or edge, enabling consistent refinement.
• Apply deformation theory to analyze how the sign of the oriented 2-plane changes under small perturbations of the constraint hyperplanes, ensuring continuity of the invariant.

Experimental results

Research questions

• RQ1 Can refined invariants be constructed for rational tropical curves with unmarked vertices of valency greater than three?
• RQ2 Does a refined count of rational tropical curves in R^m (m ≥ 3) exist under constraints that make it analogous to known planar invariants?
• RQ3 Under what conditions does the weight of a tropical curve factorize into local contributions, enabling a consistent refinement?
• RQ4 How can invariance of the refined count be proven when curve weights depend on global features rather than just local vertex data?

Key findings

• A refined invariant is constructed for plane rational tropical curves with an unmarked four-valent vertex, motivated by the correspondence with unicuspidal algebraic curves.
• The refined invariant specializes to the number of plane rational unicuspidal curves of a given degree when the formal parameter z is set to 1.
• A new refined invariant is defined for rational tropical curves in R^m (m ≥ 2) under specific constraints, generalizing planar invariants to higher dimensions.
• The invariance of the refined count is proven by verifying that certain combinatorial identities involving z^{a∧b} − z^{b∧a} hold in all possible configurations of edge directions.
• The proof relies on analyzing the orientation of 2-planes spanned by directing vectors under small deformations of the constraint hyperplanes, ensuring continuity of the sign function.
• The method successfully overcomes the difficulty of non-local weights by identifying settings where the total weight factors into independent local contributions.