Representations of Configurations and their Moduli

Abstract

This thesis develops a scheme-theoretic framework for studying configurations—collections of points and blocks with incidence relations—and their representations in algebraic geometry. Generalizing classical notions, we define configurations as triples of schemes over a base and show that the moduli functor of representations into a geometric configuration is representable by a scheme. We construct fine moduli spaces for nondegenerate representations and realizations, and study their behavior under deformation. Applications include a modern perspective on Mnëv–Sturmfels universality and connections to classical projective theorems, highlighting new interactions between configuration theory, moduli spaces, and algebraic geometry.