Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Spaces of multiscaled lines with collision

Antonios-Alexandros Robotis

Abstract

We study varieties $\mathcal{A}_n$ arising as equivariant compactifications of the space of $n$ points in $\mathbb{C}$ up to overall translation. We define $\mathcal{A}_n$ and examine its basic geometric properties before constructing an isomorphism to an augmented wonderful variety. We show that $\mathcal{A}_n$ is in a canonical way a resolution of the space $\overline{P}_n$ considered by Zahariuc, proving along the way that the resolution constructed by Zahariuc is equivalent to ours.

Spaces of multiscaled lines with collision

Abstract

We study varieties arising as equivariant compactifications of the space of points in up to overall translation. We define and examine its basic geometric properties before constructing an isomorphism to an augmented wonderful variety. We show that is in a canonical way a resolution of the space considered by Zahariuc, proving along the way that the resolution constructed by Zahariuc is equivalent to ours.
Paper Structure (20 sections, 69 theorems, 25 equations)

This paper contains 20 sections, 69 theorems, 25 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on discrete structures
  3. Level trees
  4. Partitions
  5. Matroids
  6. Subspace arrangements from partitions
  7. Multiscaled lines with collision
  8. Multiscaled lines
  9. Constructing the space
  10. Level tree stratification
  11. Symmetric group action and collision stratification
  12. Multiscaled lines as an equivariant compactification
  13. Comparison with the augmented wonderful variety
  14. Iterated blowup constructions
  15. The isomorphism
  16. ...and 5 more sections

Key Result

Theorem 1

$\mathcal{A}_n$ is a compact complex algebraic manifold of dimension $n-1$ such that $\bA^n/\bG_a \hookrightarrow \mathcal{A}_n$ is an open immersion.

Theorems & Definitions (164)

  • Theorem : \ref{['T:spaceconstruction']}
  • Theorem : \ref{['T:isothm']}
  • Theorem : \ref{['T:diagramcommutes']}
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Lemma 2.6
  • proof
  • ...and 154 more