Incidence strata of affine varieties with complex multiplicities

TL;DR

This paper constructs a family of varieties generalizing incidence strata of symmetric powers of affine varieties with complex multiplicities, analyzing their properties and singularities, and connecting to Deligne categories.

Contribution

It introduces a novel construction of incidence strata with complex multiplicities, extending classical cases, and studies their singularities and functorial properties.

Findings

Constructed a family of varieties for complex multiplicities that generalize classical incidence strata.

Verified the construction's consistency with Deligne category $Rep(S_{d})$.

Classified singularities and branching behavior for incidence strata on smooth curves.

Abstract

To each affine variety $X$ and $m_1,\ldots,m_k\in \mathbb{C}$ such that no subset of the $m_i$ add to zero, we construct a variety which for $m_1,\ldots,m_k \in \mathbb{N}$ specializes to the closed $(m_1,\ldots,m_k)$-incidence stratum of $Sym^{m_1+\ldots+m_k}X$. 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.