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Kalinin Effectivity and Wonderful Compactifications

Viatcheslav Kharlamov, Rareş Răsdeaconu

Abstract

We review the definition and main properties of Kalinin effectivity and describe methods for constructing effective spaces together with several examples. We analyze the Kalinin effectivity of wonderful compactifications and prove that the wonderful compactifications of hyperplane arrangements and of configuration spaces associated to Kalinin effective compact complex manifolds are themselves Kalinin effective. As an application, we show that the Deligne-Mumford space of real rational curves with marked points is effective. Finally, we apply Kalinin effectivity to study Smith-Thom maximality for Hilbert squares.

Kalinin Effectivity and Wonderful Compactifications

Abstract

We review the definition and main properties of Kalinin effectivity and describe methods for constructing effective spaces together with several examples. We analyze the Kalinin effectivity of wonderful compactifications and prove that the wonderful compactifications of hyperplane arrangements and of configuration spaces associated to Kalinin effective compact complex manifolds are themselves Kalinin effective. As an application, we show that the Deligne-Mumford space of real rational curves with marked points is effective. Finally, we apply Kalinin effectivity to study Smith-Thom maximality for Hilbert squares.
Paper Structure (26 sections, 54 theorems, 109 equations, 1 figure)

This paper contains 26 sections, 54 theorems, 109 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Motivation and Context
  3. Framework: Smith Theory, Kalinin Effectivity, and Conjugation Spaces
  4. Main Results and Applications
  5. Kalinin effectivity and maximality
  6. The Kalinin spectral sequence
  7. Effectivity
  8. Maximality
  9. Conjugation Spaces
  10. Examples and constructions
  11. Examples of Kalinin effective spaces
  12. Kalinin effectivity under standard constructions
  13. Effective blowing-up
  14. Constructions of maximal spaces
  15. Maximal blowing-up
  16. ...and 11 more sections

Key Result

Theorem 1

The De Concini - Procesi wonderful compactification of any arrangement of real linear subspaces in $\mathbb{P}^n$ is a conjugation space.

Figures (1)

  • Figure 1: Stretchedness for triples.

Theorems & Definitions (79)

  • Theorem 1
  • Theorem 2
  • Corollary 3
  • Theorem 4
  • Theorem 5
  • Corollary 6
  • Definition 2.1
  • Remark 2.2
  • Theorem 2.3: Kalinin, kalinin-ss
  • Proposition 2.4: Kalinin, kalinin-ss
  • ...and 69 more