k-leaky double Hurwitz descendants

Abstract

We define a new class of enumerative invariants called k-leaky double Hurwitz descendants, generalizing both descendant integrals of double ramification cycles and the k-leaky double Hurwitz numbers introduced in [CMR25]. These numbers are defined as intersection numbers of the logarithmic DR cycle against $\psi $ -classes and logarithmic classes coming from piecewise polynomials encoding fixed branch point conditions. We give a tropical graph sum formula for these new invariants, allowing us to show their piecewise polynomiality in any genus. Investigating the piecewise polynomial structure further (and restricting to genus zero for this purpose), we also show a wall-crossing formula. We also prove that in genus zero the invariants are always nonnegative and give a complete classification of the cases where they vanish.

Keywords

• 14N35
• 14T90