The Self-Projecting Grassmannian

TL;DR

This paper introduces the self-projecting Grassmannian, a new subvariety characterized by a self-duality condition, exploring its connections to classical moduli spaces and matroid theory, with computational analysis of its realization spaces.

Contribution

It defines the self-projecting Grassmannian and self-projecting matroids, linking geometric, combinatorial, and computational perspectives in a novel way.

Findings

Established the self-projecting Grassmannian as a new geometric object.

Connected self-projectivity to classical moduli spaces and matroid theory.

Performed computational exploration of realization spaces within this framework.

Abstract

We introduce the self-projecting Grassmannian, an irreducible subvariety of the Grassmannian parametrizing linear subspaces that satisfy a generalized self-duality condition. We study its relation to classical moduli spaces, such as the moduli spaces of pointed curves of genus $g$, as well as to other natural subvarieties of the Grassmannian. We further translate the self-projectivity condition in the combinatorial language of matroids, introducing self-projecting matroids, and we computationally investigate their realization spaces inside the self-projecting Grassmannian.