Universal Formulas for de Jonquières Loci and Equivariant Contact Problems

Abstract

Given a family of genus $g$ curves $f:Ψ\to S$ and a relative degree $d$ line bundle $L$ on $Ψ$ such that $f_*L$ is a vector bundle, the projectivization $\mathbb{P}(f_*L)$ is stratified by relative de Jonquières loci, which parameterize sections of $L$ with prescribed vanishing orders. We give universal formulas for the classes of these loci in terms of the relative $O(1)$ on $\mathbb{P}(f_*L)$ and tautological classes pulled back from the universal Picard stack $\mathrm{Pic}_g^d$. When $f$ is the family of lines in $\mathbb{P}^r$, we use the universal polynomials to compute $\mathrm{GL}_{r+1}$-equivariant classes of loci of hypersurfaces and complete intersections admitting lines with prescribed contact orders.