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On the Castelnuovo-Mumford regularity of subspace arrangements

Aldo Conca, Manolis C. Tsakiris

Abstract

Let $X$ be the union of $n$ generic linear subspaces of codimension $>1$ in $\mathbb{P}^d$. Improving an earlier bound due to Derksen and Sidman, we prove that the Castelnuovo-Mumford regularity of $X$ satisfies $ \operatorname{reg}(X) \le n - [n / (2d-1)]$.

On the Castelnuovo-Mumford regularity of subspace arrangements

Abstract

Let be the union of generic linear subspaces of codimension in . Improving an earlier bound due to Derksen and Sidman, we prove that the Castelnuovo-Mumford regularity of satisfies .
Paper Structure (16 sections, 22 theorems, 115 equations)

This paper contains 16 sections, 22 theorems, 115 equations.

Table of Contents

  1. Introduction
  2. Conventions
  3. Preliminaries
  4. Regularity
  5. Subspace Arrangements
  6. Flat Degenerations and Hilbert functions
  7. Proof of Theorem \ref{['thm:codim2']}
  8. First Hilbert Function Lemma
  9. Second Hilbert Function Lemma
  10. Sundials
  11. A Variant of Castelnuovo's Inequality
  12. Third Hilbert Function Lemma
  13. Saturation
  14. Finishing the Proof of Theorem \ref{['thm:codim2']}
  15. Proof of Theorem \ref{['thm:JL']}
  16. ...and 1 more sections

Key Result

Theorem 1

For $n$ generic linear spaces $X_1,\dots,X_n$ of codimension bigger than $1$ in $\mathbb{P}_k^d$, the regularity of their reduced union $X$ satisfies

Theorems & Definitions (35)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Proposition 4
  • Proposition 5
  • Definition 6
  • Proposition 7: Conca & Herzog, conca2003castelnuovo
  • Proposition 8: Derksen & Sidman, derksen2002sharp
  • Proposition 9: Derksen, derksen2007hilbert
  • Remark 10
  • ...and 25 more