Enumerating log rational curves on some toric varieties

Technical Summary: Enumerating Log Rational Curves on Some Toric Varieties

Problem Statement
This paper addresses the enumeration of genus 0, fixed-domain logarithmic Gromov-Witten invariants for smooth, projective toric varieties. Specifically, it seeks to determine the number of maps from a fixed general pointed rational curve to a toric variety $X$ that pass through the maximal number of general points and satisfy prescribed tangency multiplicities along the toric boundary. These invariants, termed "logarithmic Tevelev degrees" ($\text{logTev}^X_{\Gamma}$), were previously studied by Cela–Iribar L´opez [CIL24] using tropical geometry, leading to conjectures for specific toric varieties. The authors aim to verify or disprove these conjectures using direct algebro-geometric intersection theory rather than tropical combinatorics.

Methodology
The authors employ a direct intersection-theoretic approach on moduli spaces of "naive log quasimaps."

1. Moduli Construction: For the primary family of varieties $X_{r,s,a} = \mathbb{P}(\mathcal{O}_s \oplus \mathcal{O}(-a)) \to \mathbb{P}^r$, the authors construct a compactification $Q_{\Gamma}(X_{r,s,a})$ of the moduli space of log maps $M_{\Gamma}(X_{r,s,a})$. This space is realized as a tower of projective bundles over a product of projective lines parametrizing the positions of the boundary intersection points.
2. Naive vs. Stable: Unlike the moduli space of log stable maps, $Q_{\Gamma}$ allows for arbitrary collisions of boundary points and does not strictly enforce stability conditions, making it a "naive" space. However, it admits a simple description as a tower of projective bundles, facilitating explicit intersection calculations.
3. Transversality and Twisting: A critical technical component involves analyzing the transversality of incidence conditions. The authors define an algorithm to "twist" quasimaps that fail to be base-point-free (bpf) into maps with modified tangency data. They prove that if the intersection of incidence loci is non-empty, the maps must satisfy specific dimension constraints; otherwise, the intersection is empty.
4. Excess Intersection: In the case of blow-ups of $\mathbb{P}^r$, the authors utilize the excess intersection formula to compute contributions from loci where the intersection number fails to be enumerative (i.e., where the intersection is not reduced or supported on the expected locus).

Key Results

• Complete Determination for Projective Bundles: The authors prove a conjecture of Cela–Iribar L´opez [CIL24, Conjecture 14] regarding the logarithmic Tevelev degrees for $X_{r,s,a} = \mathbb{P}(\mathcal{O}_s \oplus \mathcal{O}(-a))$. They derive an explicit formula for $\text{logTev}^{X_{r,s,a}}_{\Gamma}$ (Theorem 1.3.1 / Theorem 4.2.1).

  • The formula is non-zero only if specific inequalities regarding the tangency multiplicities $m_j$ and the number of marked points $n$ are satisfied (e.g., $m_j \leq n-1$).
  • When these conditions hold, the invariant is given by a product of factorials, a binomial coefficient involving a derived integer $k_0$, and a power of $a$.
  • If the inequalities fail, the invariant is proven to be zero.
  • This result generalizes previous computations for Hirzebruch surfaces ($r=s=1$) and provides a complete enumerative answer for this family of toric varieties.

• Disproof of Conjecture for Blow-ups: The authors disprove [CIL24, Conjecture 15], which predicted the logarithmic Tevelev degrees for the blow-up of $\mathbb{P}^r$ at $r$ torus-fixed points.

  • Focusing on the blow-up of $\mathbb{P}^2$ at 2 points, the authors demonstrate that the predicted value (derived from a tautological intersection number on their naive moduli space) does not always equal the actual geometric count.
  • They construct a specific counterexample (Theorem 5.2.2) where the actual invariant is non-zero but differs from the conjectured value.
  • The discrepancy arises because the intersection number on the naive moduli space includes contributions from "excess" loci (where sections vanish identically) that are not enumerative. The authors explicitly compute these excess contributions using the excess intersection formula, showing they do not vanish in all cases.

Significance and Claims
The paper claims to resolve the enumerative problem for a broad class of toric varieties ($X_{r,s,a}$) using direct algebro-geometric methods, bypassing the combinatorial complexity of tropical correspondence theorems.

• Enumerativity: The authors highlight that while fixed-domain Gromov-Witten invariants for non-logarithmic settings often fail to be enumerative (as noted in [BLL+24]), the logarithmic setting considered here yields enumerative counts for the varieties $X_{r,s,a}$.
• Methodological Contribution: The work establishes that "naive log quasimaps" provide a viable framework for explicit calculation, provided one carefully analyzes the boundary behavior and excess intersection contributions.
• Correction of Conjectures: The paper provides a definitive correction to the tropical-based predictions for blow-ups of projective spaces, demonstrating that the tropical correspondence for fixed-domain invariants requires more subtle algebro-geometric verification than previously assumed.

The authors do not claim to have solved the problem for all toric varieties or higher genus, but rather provide a complete solution for the specific family $X_{r,s,a}$ and a counterexample to the general conjecture for blow-ups.