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A Second Main Theorem for Entire Curves Intersecting Three Conics

Lei Hou, Dinh Tuan Huynh, Joël Merker, Song-Yan Xie

TL;DR

The paper proves an effective Second Main Theorem for entire curves in P^2 intersecting a generic configuration of three conics, establishing a concrete bound T_f(r) ≤ 5 Σ_i N_f^{[1]}(r, C_i) + o(T_f(r)). It combines invariant logarithmic 2-jet differential methods with Slanted Vector Field techniques, the log Demailly–Semple tower, and Bogomolov-type vanishing to control the base locus and derive explicit constants. The result advances hyperbolicity questions for plane complements with low-degree components by providing an explicit, near-optimal constant obtained through a blend of Riemann–Roch computations, Wronskian interpretations, and computer-assisted vanishing lemmas. These contributions deepen the toolkit for effective SMTs in low-degree, nonlinear configurations and illustrate how higher-jet techniques can tighten bounds in complex hyperbolicity problems.

Abstract

We establish a Second Main Theorem for entire holomorphic curves $f: \mathbb{C} \to \mathbb{P}^2$ intersecting a generic configuration of three conics $\mathcal{C}_1, \mathcal{C}_2, \mathcal{C}_3$ in the complex projective plane $\mathbb{P}^2$. Using invariant logarithmic $2$-jet differentials with negative twists, we prove the estimate \[ T_f(r) \leqslant 5 \sum_{i=1}^3 N_f^{[1]}(r, \mathcal{C}_i) + o\big(T_f(r)\big)\quad\parallel, \] where $T_f(r)$ is the Nevanlinna characteristic function, and $N_f^{[1]}(r, \mathcal{C}_i)$ is the 1-truncated counting function.

A Second Main Theorem for Entire Curves Intersecting Three Conics

TL;DR

The paper proves an effective Second Main Theorem for entire curves in P^2 intersecting a generic configuration of three conics, establishing a concrete bound T_f(r) ≤ 5 Σ_i N_f^{[1]}(r, C_i) + o(T_f(r)). It combines invariant logarithmic 2-jet differential methods with Slanted Vector Field techniques, the log Demailly–Semple tower, and Bogomolov-type vanishing to control the base locus and derive explicit constants. The result advances hyperbolicity questions for plane complements with low-degree components by providing an explicit, near-optimal constant obtained through a blend of Riemann–Roch computations, Wronskian interpretations, and computer-assisted vanishing lemmas. These contributions deepen the toolkit for effective SMTs in low-degree, nonlinear configurations and illustrate how higher-jet techniques can tighten bounds in complex hyperbolicity problems.

Abstract

We establish a Second Main Theorem for entire holomorphic curves intersecting a generic configuration of three conics in the complex projective plane . Using invariant logarithmic -jet differentials with negative twists, we prove the estimate \[ T_f(r) \leqslant 5 \sum_{i=1}^3 N_f^{[1]}(r, \mathcal{C}_i) + o\big(T_f(r)\big)\quad\parallel, \] where is the Nevanlinna characteristic function, and is the 1-truncated counting function.

Paper Structure

This paper contains 31 sections, 21 theorems, 190 equations, 1 table.

Table of Contents

  1. Introduction
  2. Motivation
  3. Two Key Theorems
  4. Our First Result
  5. Proof of Theorem \ref{['smt for three components, case large degrees']}
  6. Our Second, Main Result: Second Main Theorem for Three Conics.
  7. Roadmap for Theorem \ref{['main theorem 3 conics']}: Towards an Effective Finite Constant $c$
  8. New Vanishing Results and Proof Strategy
  9. Proof Blueprint of Theorem \ref{['main theorem 3 conics']} with an Effective Constant $c=5$
  10. A Challenge
  11. Structure of the Paper
  12. Preliminaries
  13. Logarithmic Jet Bundles and Invariant Differential Forms
  14. Logarithmic Demailly-Semple Tower
  15. A Bogomolov-type Vanishing Theorem
  16. ...and 16 more sections

Key Result

Theorem 1.1

Let $\mathcal{C}=\sum_{i=1}^q\mathcal{C}_i$ be a normal crossings divisor in $\mathbb{P}^2$, where each $\mathcal{C}_i$ is a smooth algebraic curve of degree $d_i$ with $d=\sum_{i=1}^q d_i\geqslant 4$. Let $f:\mathbb{C}\rightarrow\mathbb{P}^2$ be an algebraically nondegenerate entire holomorphic cur

Theorems & Definitions (43)

  • Theorem 1.1: McQuillan Mcquillan1998
  • Theorem 1.2: Huynh-Vu-Xie-2017
  • Theorem 1.3
  • Remark 1.4
  • proof : Proof of Theorem \ref{['smt for three components, case large degrees']} assuming \ref{['key 1-jet differential existence']}
  • Theorem 1.6: Second Main Theorem for Three Conics
  • Remark 1.7
  • Remark 1.8
  • Remark 1.9
  • Conjecture 1.10
  • ...and 33 more