[Paper Review] The Coolidge-Nagata conjecture holds for curves with more than four cusps

Karol Palka|arXiv (Cornell University)|Feb 16, 2012

Algebraic Geometry and Number Theory8 references3 citations

TL;DR

This paper proves the Coolidge-Nagata conjecture for rational cuspidal curves in the complex projective plane with more than four cusps, showing such curves are rectifiable via a birational automorphism of ℙ². Using embedded resolution of singularities and analysis of the exceptional divisor's tree structure, the authors establish that non-rectifiable curves can have at most nine maximal twigs in their resolution tree, thereby confirming the conjecture in the multi-cuspidal case.

ABSTRACT

Let E be a plane rational curve defined over complex numbers which has only locally irreducible singularities. The Coolidge-Nagata conjecture states that E is rectifiable, i.e. it can be transformed into a line by a birational automorphism of the plane. We show that if it is not rectifiable then the tree of the exceptional divisor for its minimal embedded resolution of singularities has at most nine maximal twigs. This settles the conjecture in case E has more than four singular points.

Motivation & Objective

• To resolve the Coolidge-Nagata conjecture for rational cuspidal curves in ℙ² with more than four cusps.
• To determine structural constraints on the minimal embedded resolution of singularities for non-rectifiable rational cuspidal curves.
• To show that if such a curve is non-rectifiable, its exceptional divisor tree has at most nine maximal twigs.
• To establish that the conjecture holds universally for curves with more than four cusps, regardless of their existence.

Proposed method

• Analyzes the minimal embedded resolution of singularities of a rational cuspidal curve E ⊆ ℙ².
• Applies Zariski decomposition to the canonical divisor K + E and studies the Iitaka-Kodaira dimension κ(K + E).
• Uses the bark of the exceptional divisor, defined via the formula Bk T · T₀ = β_T(T₀) - 2 for components T₀ of maximal twigs.
• Applies Noether's formula and discriminant invariants d(T) for rational chains and trees to constrain possible configurations.
• Employs characteristic pair sequences of type *(n,k) to model the resolution process at singular points and compute invariants like M(q) and I(q).
• Derives contradictions via modular arithmetic (mod 5, mod 11) when assuming ten or more maximal twigs, particularly in the four- and five-cusp cases.

Experimental results

Research questions

• RQ1 Can a rational cuspidal curve in ℙ² with more than four cusps fail to be rectifiable by a birational automorphism of ℙ²?
• RQ2 What structural limitations exist on the exceptional divisor tree of a non-rectifiable rational cuspidal curve?
• RQ3 Is the number of maximal twigs in the resolution tree of a non-rectifiable rational cuspidal curve bounded?
• RQ4 Does the existence of a curve with more than four cusps imply rectifiability, regardless of whether such curves exist?
• RQ5 Can the resolution process at singular points be modeled using characteristic pair sequences of type *(n,k) to derive contradictions under high twig counts?

Key findings

• If a rational cuspidal curve in ℙ² has more than four cusps, it is necessarily rectifiable by a birational automorphism of ℙ².
• Any non-rectifiable rational cuspidal curve in ℙ² has an exceptional divisor tree with at most nine maximal twigs in its minimal embedded resolution.
• The case of five cusps with ten maximal twigs leads to a contradiction via modular arithmetic on invariants M(q) and I(q), ruling out such configurations.
• The case of four cusps with ten maximal twigs also leads to a contradiction, confirming the upper bound of nine maximal twigs for non-rectifiable curves.
• The analysis of characteristic pair sequences of type *(n,k) allows the derivation of precise formulas for the self-intersection and length of resolution chains, enabling contradiction-based exclusion of high-twig configurations.
• The Noether formula K² + #D = 10, combined with invariants K·(K+D) and δ(D), is instrumental in deriving the contradiction when assuming too many twigs.