Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Self-Projecting Grassmannian

Alheydis Geiger, Francesca Zaffalon

TL;DR

The paper introduces the self-projecting Grassmannian $\mathrm{SGr}(k,n)$ as the Zariski closure of subspaces $V\in\mathrm{Gr}(k,n)$ for which $V\,\mathrm{diag}(\lambda)\,V^t=0$ for some diagonal $\lambda$, unifying self-duality and torus orbit considerations. It establishes irreducibility and a precise dimension formula when $2k\le n\le {k+1\choose2}$, and shows $\mathrm{SGr}(k,n)=\mathrm{Gr}(k,n)$ for $n>{k+1\choose2}$ while $\mathrm{SGr}(k,n)=\emptyset$ for $n<2k$, with every self-projecting space lying in some orthogonal Grassmannian. The work connects geometric configurations to moduli spaces via birational equivalences (e.g., $\mathrm{SGr}(4,9)^{\circ}/(\mathbb{C}^*)^9$ with $\mathcal{M}_{1,10}$ and $\mathrm{SGr}(5,13)^{\circ}$ with $\mathcal{M}_{5,13}$), and translates the construction to the Combinatorial realm by introducing self-projecting matroids, analyzing their realization spaces, and providing extensive computational data. The paper further explores the real and positive aspects, including real orthogonal Grassmannians, totally non-negative realizations, and self-projecting positroids, highlighting both expected and surprising phenomena and laying a computational foundation for future study of positivity and tropical aspects.

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.

The Self-Projecting Grassmannian

TL;DR

The paper introduces the self-projecting Grassmannian as the Zariski closure of subspaces for which for some diagonal , unifying self-duality and torus orbit considerations. It establishes irreducibility and a precise dimension formula when , and shows for while for , with every self-projecting space lying in some orthogonal Grassmannian. The work connects geometric configurations to moduli spaces via birational equivalences (e.g., with and with ), and translates the construction to the Combinatorial realm by introducing self-projecting matroids, analyzing their realization spaces, and providing extensive computational data. The paper further explores the real and positive aspects, including real orthogonal Grassmannians, totally non-negative realizations, and self-projecting positroids, highlighting both expected and surprising phenomena and laying a computational foundation for future study of positivity and tropical aspects.

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 , 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.

Paper Structure

This paper contains 18 sections, 24 theorems, 22 equations, 5 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. The self-projecting Grassmannian
  3. Definitions
  4. The orthogonal Grassmannian
  5. Point configurations $X(k,n)$
  6. Associated point configurations
  7. Point configurations on hypersurfaces $X_{r,d^m,n}$ and beyond
  8. Geometric interpretations
  9. 9 self-projecting points in $\mathbb{P}^3$
  10. 10 self-projecting points in $\mathbb{P}^3$
  11. 13 self-projecting points in $\mathbb{P}^4$
  12. Self-projecting matroids and their realization spaces
  13. Self-projecting matroids
  14. Self-projecting matroid realization spaces
  15. The computations, the code and the database
  16. ...and 3 more sections

Key Result

Proposition 2.2

The self-projecting Grassmannian can be defined in dual Stiefel coordinates as the variety $\mathcal{V}(I^{\mathop{\mathrm{sd}}\nolimits}_{k,n})$ with where $X$ denotes the matrix with entries $X_{ij} = x_{(i,j)}$.

Theorems & Definitions (58)

  • Definition 2.1
  • Proposition 2.2
  • Definition 2.3
  • Corollary 2.4
  • Proposition 2.5
  • proof
  • Definition 2.6
  • Proposition 2.7
  • proof
  • Remark 2.8
  • ...and 48 more