Incidence strata of affine varieties with complex multiplicities
Hunter Spink, Dennis Tseng

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 and such that no subset of the add to zero, we construct a variety which for specializes to the closed -incidence stratum of . These fit into a finite-type family, which is functorial in , and which is topologically a family of -weighted configuration spaces. We verify our construction agrees with an analogous construction in the Deligne category for . 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.
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Advanced Algebra and Geometry · Polynomial and algebraic computation
