Generalizations of tropical Tevelev degrees

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.

💡 Research Summary

The paper expands the recently introduced tropical Tevelev degrees, which count the degree of a natural morphism between tropical admissible‑cover spaces and tropical moduli of curves, by adding a new integer parameter ℓ and by allowing arbitrary ramification profiles. In the classical setting (d = g + 1, n = g + 3) the tropical degree equals 2g, matching the algebraic Tevelev degree. The authors consider covers of degree d = g + 1 + ℓ with n = g + 3 + 2ℓ marked points, a choice that keeps source and target dimensions equal (both 5g + 4ℓ).

For ℓ > 0 the degree does not change: the combinatorial construction from Cavaliere–Dawson still applies. Each tropical cover possesses a unique “active path” on the source graph; along this path one performs a sequence of U (increase) and D (decrease) operations. The sequence length is g‑1 and each step has two choices, giving 2^{g‑1} possibilities. Together with the fixed degree‑2 genus‑one core and the marked‑tree fragment, the total number of covers is 2g, so the tropical Tevelev degree remains 2g (Theorem 1.1).

When ℓ < 0 the degree d is smaller, and the Riemann–Hurwitz condition forces the active path never to drop below a certain height. Consequently many U/D words become forbidden. The authors compute the exact loss as a sum of binomial differences: \