Tevelev degrees and Hurwitz moduli spaces

TL;DR

This paper links Tevelev degrees from scattering amplitudes to Hurwitz moduli spaces, deriving a recursion and exact formulas that connect algebraic geometry, combinatorics, and classical enumerative geometry.

Contribution

It introduces a new recursion for Tevelev degrees using excess intersection theory and provides exact solutions, including a novel two-parameter generalization.

Findings

Derived a simple recursion for Tevelev degrees.

Connected Tevelev degrees to classical counts of linear series.

Solved a new two-parameter generalization and related combinatorial counts.

Abstract

We interpret the degrees which arise in Tevelev's study of scattering amplitudes in terms of moduli spaces of Hurwitz covers. Via excess intersection theory, the boundary geometry of the Hurwitz moduli space yields a simple recursion for the Tevelev degrees (together with their natural two parameter generalization). We find exact solutions which specialize to Tevelev's formula in his cases and connect to the projective geometry of lines and Castelnuovo's classical count of linear series in other cases. For almost all values, the calculation of the two parameter generalization of the Tevelev degree is new. A related count of refined Dyck paths is solved along the way.