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The Cayley-Bacharach property and the Levinson-Ullery conjecture

Ngoc Long Le, Tran N. K. Linh

TL;DR

The paper addresses the geometric structure of finite point sets in projective space with the Cayley–Bacharach property (CBP) and the Levinson–Ullery plane-configuration conjecture. It develops algebraic tools—Hilbert functions, the canonical module, and equivalences for CBP(r)—and introduces plane configurations to capture the geometric decompositions of CBP-satisfied sets. The main result proves the LU conjecture for the case d=4 and all r≥1, using an induction on r and a detailed case analysis guided by CBP-induced bounds and plane-configuration decompositions. This advances understanding of how CBP constrains point configurations and informs related topics in algebraic geometry and combinatorial geometry.

Abstract

In this paper, we study the geometric configurations of a finite set of points having the Cayley-Bacharach property in the $n$-dimensional projective space $\bbP^n$. Our main contribution is the proof of the Levinson-Ullery conjecture for the previously unsolved case where $d=4$ and $r\ge 1$.

The Cayley-Bacharach property and the Levinson-Ullery conjecture

TL;DR

The paper addresses the geometric structure of finite point sets in projective space with the Cayley–Bacharach property (CBP) and the Levinson–Ullery plane-configuration conjecture. It develops algebraic tools—Hilbert functions, the canonical module, and equivalences for CBP(r)—and introduces plane configurations to capture the geometric decompositions of CBP-satisfied sets. The main result proves the LU conjecture for the case d=4 and all r≥1, using an induction on r and a detailed case analysis guided by CBP-induced bounds and plane-configuration decompositions. This advances understanding of how CBP constrains point configurations and informs related topics in algebraic geometry and combinatorial geometry.

Abstract

In this paper, we study the geometric configurations of a finite set of points having the Cayley-Bacharach property in the -dimensional projective space . Our main contribution is the proof of the Levinson-Ullery conjecture for the previously unsolved case where and .

Paper Structure

This paper contains 3 sections, 8 theorems, 7 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Proof of main theorem

Key Result

Theorem 1.1

Let ${\mathbb{X}}\subseteq {\mathbb{P}}^n$ be a finite set of points having $\mathop{\rm CBP}\nolimits(r)$. If then ${\mathbb{X}}$ lies on a line.

Theorems & Definitions (16)

  • Theorem 1.1
  • Conjecture 1.2: Levinson-Ullery
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1
  • Remark 2.2
  • Proposition 2.3
  • Corollary 2.4
  • proof
  • Definition 2.5
  • ...and 6 more