Generalizations of tropical Tevelev degrees

TL;DR

This work extends tropical Tevelev theory by introducing an integer parameter $\ell$ that varies the degree and marked-point data to $d=g+1+\ell$ and $n=g+3+2\ell$, and by defining tropical generalized Tevelev degrees for ramification profiles $\mu_1,\dots,\mu_k$. It proves that for $\ell>0$ the tropical Tevelev degree remains $2^g$, while for $\ell<0$ a explicit deficit is described in closed form, aligning with algebraic expectations under specialization. It then provides a tropical counterpart to the generalized Tevelev degrees of Cela and Lian, giving closed-form combinatorial expressions that recover algebraic values when $\ell$ is negative and ramification profiles are imposed. The approach relies on a detailed tropical admissible-cover framework, lattice-path/grid combinatorics, and a robust correspondence with algebraic results, yielding a uniform, computable program for tropical intersection numbers on moduli spaces. Overall, the paper broadens the scope of tropical Tevelev theory and clarifies how degree variation and ramification interact in tropical intersection theory, with potential applications to tropical Hurwitz theory and moduli-space computations.

Abstract

We study tropical Tevelev degrees arising from maps between certain tropical moduli spaces of curves. Building on work of Dawson and Cavalieri, who defined and computed tropical Tevelev degrees in the case of degree $d = g+1$ and $n = g+3$ marked points, we extend the theory by introducing an additional integer parameter $\ell$. In our framework the curve degree and number of marked points vary as $d = g + 1 + \ell$ and $n = g + 3 + 2\ell$, and we analyze the resulting tropical Tevelev degrees for both positive and negative values of $\ell$. This tropicalizes results of Cela, Pandharipande, and Schmitt on algebraic Tevelev degrees. We then further broaden the framework by introducing generalized tropical Tevelev degrees, providing the tropical counterpart to the generalized Tevelev degrees studied by Cela and Lian. These results establish a wider set of computational and structural patterns for intersection calculations on tropical moduli spaces and reveal new behavior beyond the classical setting.

Figures (36)

• Figure 1: The graphs $\overline{\Gamma}, \overline{T}$ defining the chosen point $p$ of $\mathcal{M}_{g,n}^{\mathtt{trop}}\times \mathcal{M}_{0,n}^{\mathtt{trop}}$. This figure is reproduced from troptev.
• Figure 2: Two possible ways to add genus $U$, on left, and $D$, on right. We omit from the picture ends of degree $1$ to avoid clutter. The active path is thickened. Note that adding $U$ increases degree of active edge from right to left, while adding $D$ decreases the degree. This figure is reproduced from troptev.
• Figure 3: Marked fragments that attach to the horizontal edge of $\tilde{T}$ to obtain the base graph $T$. We denote by $F_j^-$ the connected component that contains the marks with the lowest indices, and $F_j^+$ the one containing the highest labels. This figure is reproduced from troptev.
• Figure 4: The grid of solutions constructed for counting covers when $g=3, n=6$. Adapted from troptev.
• Figure 5: The point $p$ in $\mathcal{M}_{1,6}^{\mathtt{trop}} \times \mathcal{M}_{0,6}^{\mathtt{trop}}$. We have $x_1<<x_2<<x_3<<x_4<<x_5<<x_6<<L_1<<L_2<<L_3$.

Theorems & Definitions (20)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 2.1
• Definition 2.2
• Theorem 2.3: troptev
• Theorem 2.4: troptev
• Definition 3.1
• Remark 3.2
• Claim 4.1