[Paper Review] On the number of Galois points for a plane curve in positive characterisitic, IV

Satoru Fukasawa|arXiv (Cornell University)|Nov 16, 2010

Algebraic Geometry and Number Theory11 citations

TL;DR

This paper completes the classification of smooth plane curves by determining the exact numbers of Galois points on the curve (δ(C)) and outside it (δ'(C)) in all remaining open cases. It provides new characterizations of the Fermat curve and Klein quartic curve using δ'(C), establishing a complete invariant classification based on Galois point counts in positive characteristic.

ABSTRACT

Let $C$ be a smooth plane curve. A point $P$ in the projective plane is said to be Galois with respect to $C$ if the function field extension induced from the point projection from $P$ is Galois. We denote by $\delta(C)$ (resp. $\delta'(C)$) the number of Galois points contained in $C$ (resp. in $\mathbb P^2 \setminus C$). In this article, we determine the numbers $\delta(C)$ and $\delta'(C)$ in any remaining open cases. Summarizing results obtained by now, we will have a complete classification theorem of smooth plane curves by the number $\delta(C)$ or $\delta'(C)$. In particular, we give new characterizations of Fermat curve and Klein quartic curve by the number $\delta'(C)$.

Motivation & Objective

• To resolve the final open cases in the classification of smooth plane curves based on the number of Galois points.
• To determine the precise values of δ(C) and δ'(C) for all smooth plane curves in positive characteristic.
• To establish new algebraic invariants for curve classification using δ'(C), particularly for special curves like the Fermat and Klein quartic.
• To complete the program of classifying smooth plane curves by their Galois point counts, achieving a comprehensive invariant classification.

Proposed method

• Analyzing the function field extension induced by point projection from a Galois point to classify the Galois group structure.
• Using geometric and algebraic techniques in positive characteristic to study the ramification and Galois structure of the projection morphism.
• Applying results from algebraic geometry over finite fields to constrain possible values of δ(C) and δ'(C).
• Leveraging known classification results for special curves (e.g., Fermat, Klein) to identify distinguishing invariants via δ'(C).
• Employing the theory of wild ramification and automorphism groups to analyze the Galois action on the function field.
• Combining local and global invariants to resolve previously open cases in the classification program.

Experimental results

Research questions

• RQ1 What are the exact values of δ(C) and δ'(C) for all smooth plane curves in positive characteristic, including previously unresolved cases?
• RQ2 Can the Fermat curve and Klein quartic curve be uniquely characterized by the value of δ'(C) in positive characteristic?
• RQ3 How do the numbers δ(C) and δ'(C) serve as complete invariants for classifying smooth plane curves?
• RQ4 What structural properties of the function field extension from point projection determine the Galois condition in positive characteristic?
• RQ5 Which configurations of Galois points are geometrically and arithmetically possible for smooth plane curves in positive characteristic?

Key findings

• The paper resolves all remaining open cases in the classification of smooth plane curves by δ(C) and δ'(C), completing the program.
• It establishes that δ'(C) uniquely characterizes the Fermat curve among smooth plane curves in positive characteristic.
• It provides a new characterization of the Klein quartic curve via its δ'(C) value, distinguishing it from other curves.
• The complete classification of smooth plane curves is now achieved using δ(C) or δ'(C) as invariants.
• The values of δ(C) and δ'(C) are fully determined for all smooth plane curves in positive characteristic.
• The results confirm that the number of Galois points outside the curve (δ'(C)) serves as a powerful invariant for curve classification.