Tevelev degrees and Hurwitz moduli spaces

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.

arXiv comment34 pages, results about refined Dyck path counting added

References in corpus (8)

• Hurwitz numbers and intersections on moduli spaces of curves
• Gromov-Witten theory, Hurwitz numbers, and Matrix models, I
• Double ramification cycles on the moduli spaces of curves
• A descendent relation in genus 2
• Constructions of nontautological classes on moduli spaces of curves
• Nontautological Bielliptic Cycles
• Scattering amplitudes of stable curves
• Tevelev degrees in Gromov-Witten theory