Elliptic curve counting in toric threefolds: virtual, enumerative, and tropical

Technical Summary: Elliptic Curve Counting in Toric Threefolds: Virtual, Enumerative, and Tropical

Problem Statement
This paper addresses the enumerative geometry of genus 1 (elliptic) curves in smooth toric threefolds $X$. Specifically, it investigates the relationship between two distinct counting invariants for a fixed curve class $\beta$ and cohomology conditions $\phi$:

1. Enumerative Invariants ($W_{X,\beta}(\phi)$): Integer counts of genus 1 curves passing through fixed curve and point conditions, defined via "well-spaced" tropical curves.
2. Virtual Invariants ($L_{X,\beta}(\phi)$): Logarithmic Gromov–Witten invariants associated with the pair $(X, \partial X)$, where $\partial X$ is the toric boundary.

The central problem is that logarithmic Gromov–Witten invariants are generally not enumerative; they include contributions from "superabundant" tropical curves (those with deformation spaces larger than expected) and boundary strata that do not correspond to actual algebraic curves in the enumerative sense. The paper seeks an explicit formula relating the virtual count to the enumerative count, generalizing the Getzler–Pandharipande relation for $\mathbb{P}^3$ to arbitrary toric threefolds relative to their boundary.

Methodology
The author employs a comparative approach using tropical geometry and logarithmic degeneration techniques:

• Tropical Correspondence: The enumerative invariant $W_{X,\beta}(\phi)$ is computed via a tropical correspondence theorem (established in prior work with Cela [14]) as a weighted sum over "well-spaced" tropical curves in $\mathbb{R}^3$. A tropical curve is well-spaced if, for every plane containing its circuit, the neighborhood of the circuit is not contained in the plane, or the curve leaves the plane via at least three edges at the minimum distance.
• Logarithmic Degeneration: The virtual invariant $L_{X,\beta}(\phi)$ is analyzed using the decomposition theorem for logarithmic Gromov–Witten invariants. This decomposes the invariant into a sum over tropical types, where contributions are reduced to vertex contributions via expanded degenerations.
• Stratification Analysis: The paper compares the moduli spaces of logarithmic stable maps and well-spaced logarithmic curves. It identifies that while these spaces share strata indexed by non-superabundant types, they differ significantly for superabundant types (excess dimension $e(\gamma) > 0$).
• Case-by-Case Analysis of Excess Dimensions: The author systematically analyzes tropical types based on their excess dimension $e(\gamma)$:
  • $e(\gamma) = 0$: Non-superabundant curves.
  • $e(\gamma) = 1$: Curves where the circuit lies in a plane but the neighborhood does not.
  • $e(\gamma) = 2$: Curves where the circuit lies in a line.
  • $e(\gamma) = 3$: Curves where the circuit is contracted to a single genus 1 vertex.

Key Contributions and Results

1. The Logarithmic Getzler–Pandharipande Relation (Theorem A):
   The main result establishes an explicit formula relating the enumerative and virtual invariants:
   $$W_{X,\beta}(\phi) = L_{X,\beta}(\phi) + E_1 + \frac{1}{24}E_0$$

   • $E_1$: A correction term accounting for non-rigid, well-spaced tropical curves of excess dimension 1. These contribute to the enumerative count but vanish in the logarithmic count.
   • $E_0$: A correction term derived from genus 0 tropical curves. It is a weighted sum over genus 0 types $\gamma$ and vertices $v$, involving the square of the wedge product of edge directions ($|u_1 \wedge u_2|^2$) and the tropical multiplicity $M_\gamma$. This term accounts for the difference arising from genus 1 vertices contracted to a point (excess dimension 3).

2. Vanishing and Contribution Analysis:

   • Non-superabundant types ($e=0$): The contributions to both $W$ and $L$ coincide.
   • Excess dimension 1 ($e=1$): The logarithmic contribution vanishes (Lemma 4.3.1) because the gluing conditions around the loop become trivial in the third direction, leading to a vanishing virtual class. However, well-spaced curves of this type can contribute to $W$.
   • Excess dimension 2 ($e=2$): The paper shows that rigid tropical types of this excess dimension (where the circuit lies in a line) do not contribute to the well-spaced count unless specific geometric conditions are met, while the logarithmic count vanishes for these types.
   • Excess dimension 3 ($e=3$): The logarithmic contribution is non-zero and equals $-\frac{1}{24}E_0$. The well-spaced contribution vanishes because such curves are not well-spaced under general conditions.

3. Inequality for $\mathbb{P}^3$ (Corollary B):
   Applying the main theorem to $X = \mathbb{P}^3$, the paper derives an inequality between the ordinary Gromov–Witten invariants ($GW^1_{a,b}$) and the logarithmic invariants ($L_{a,b}$):
   $$(d!)^4 GW^1_{a,b} \geq L_{a,b}$$
   The inequality is strict for sufficiently large degrees, specifically for $d \geq 3$ when considering all line conditions, or for $d \geq 4$ when considering all point conditions. This demonstrates that the ordinary Gromov–Witten invariants strictly exceed the logarithmic invariants relative to the toric boundary once the degree is high enough.

4. Computational Framework:
   The paper provides a method for calculating the correction terms using floor diagrams.

   • $E_0$ is computed by modifying the floor diagram methods of Brugallé–Mikhalkin to include vertex weights derived from the $E_0$ formula.
   • Examples are provided for low degrees ($d=3, 4$) and specific conditions, explicitly calculating the difference between the invariants.

Significance and Claims
The paper claims to provide a "logarithmic analogue" of the Getzler–Pandharipande relation for elliptic curves in $\mathbb{P}^3$, extending it to all toric threefolds relative to their toric boundary.

• Unification: It unifies logarithmic Gromov–Witten theory, higher double ramification cycles, and tropical enumeration.
• Realizability: It demonstrates how Speyer's well-spacedness condition, originally a tool for realizability, can be used to produce combinatorial formulas that correct virtual counts to obtain enumerative integers.
• Independence: The result is independent of the specific combinatorial multiplicities found in prior work [14]; rather, it relies on the structural comparison of moduli spaces.
• Limitations: The paper acknowledges that while it provides a theoretical framework and explicit formulas for the correction terms, finding efficient recursion relations or "floor decomposed" constraints for higher genus curves in general toric varieties remains an open challenge due to the complexity of superabundant deformations.

The work does not propose new experimental applications but rather offers a rigorous theoretical bridge between virtual and enumerative geometries, providing concrete methods to compute the discrepancy between them in the first non-trivial case (genus 1 in dimension 3).