Incidence strata of affine varieties with complex multiplicities

Abstract

To each affine variety $X$ and $m_1,\dots ,m_k\in \mathbb{C}$ such that no subset of the $m_i$ add to zero, we construct a variety which for $m_1,\dots ,m_k\in \mathbb{N}$ specializes to the closed $(m_1,\dots ,m_k)$-incidence stratum of $Sy{m}^{m_1+\dots +m_k}X$. These fit into a finite-type family, which is functorial in $X$, and which is topologically a family of $\mathbb{C}$-weighted configuration spaces. We verify our construction agrees with an analogous construction in the Deligne category $Rep(S_d)$ for $d\in \mathbb{C}$. We next classify the singularity locus and branching behaviour of colored incidence strata for arbitrary smooth curves. As an application, we negatively answer a question of Farb and Wolfson concerning the existence of an isomorphism between two natural moduli spaces.

27 pages, comments welcome!