Abstract

 For (X|D) a smooth pair, the Gromov—Witten theory of the root stacks of X along D with parameter r is a well-studied way of counting maps to X with given tangency to D. By involved calculations, Tseng and You have shown that the resulting invariants are polynomial in r of degree bounded in terms of the genus g of the domain curves. I will show a geometric way to understand this polynomiality by an explicit description of the moduli space of twisted maps to the ''universal pair"- a space which controls the enumerative geometry of any smooth pair (X|D). It turns out this space is built from a distinguished main component, along with abelian variety fibrations over boundary strata of this component arising from Jacobians of curves. The irreducible components of this space then correspond to certain combinatorial/tropical data, and each contributes a single monomial to the polynomial discovered by Tseng and You.