Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Low degree points on singular plane curves

Zachary R. Canale, Nathan Chen, Zoe Curewitz, Jacob A. Daum, Karina Dovgodko, Carlos F. Santiago-Calderón, Shiv R. Yajnik

Abstract

The purpose of this paper is to study low degree points on plane curves. We prove results analogous to those of Debarre and Klassen for singular plane curves with a finite number $δ$ of ordinary nodes/cusps, where $δ$ is bounded from above by a quadratic function in the degree of the plane curve.

Low degree points on singular plane curves

Abstract

The purpose of this paper is to study low degree points on plane curves. We prove results analogous to those of Debarre and Klassen for singular plane curves with a finite number of ordinary nodes/cusps, where is bounded from above by a quadratic function in the degree of the plane curve.
Paper Structure (5 sections, 13 theorems, 24 equations)

This paper contains 5 sections, 13 theorems, 24 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Nonexistence of abelian subvarieties
  4. An asymptotically sharp example
  5. Closing remarks

Key Result

Theorem 1.1

Let $C \subset \mathbb{P}^{2}$ be a plane curve over a number field $K$ with $\delta$ ordinary nodes and/or cusps. Suppose one of the following conditions holds: Then $C$ only contains finitely many points of degree $\leq d-3$, and all but finitely many points of $C$ of degree $\leq d-1$ arise as the intersection of $C$ with lines in $\mathbb{P}^2$ defined over $K$.

Theorems & Definitions (19)

  • Theorem 1.1
  • Theorem 2.2: Faltings91
  • Theorem 2.3: CK91, Theorems 2.1, 2.3, and 2.4(ii)
  • Proposition 2.4
  • proof
  • Lemma 2.5: CK91correction, Main Lemma
  • Lemma 2.6: CK91, Lemma 1.2
  • Proposition 2.7
  • proof
  • Proposition 2.8
  • ...and 9 more