Cuspidal curves, minimal models and Zaidenberg's finiteness conjecture

TL;DR

This paper proves that complex rational cuspidal curves in the projective plane have at most six cusps and provides an effective version of Zaidenberg's Finiteness Conjecture by analyzing the Minimal Model Program.

Contribution

It establishes an upper bound on the number of cusps for rational cuspidal curves and proves an effective form of Zaidenberg's Finiteness Conjecture using the Minimal Model Program.

Findings

Cuspidal curves have at most six cusps.

Effective bounds on Picard rank and boundary components.

Proof of Zaidenberg's Finiteness Conjecture in this context.

Abstract

Let $E\subseteq \mathbb{P}^2$ be a complex rational cuspidal curve and let $(X,D)\to (\mathbb{P}^2,E)$ be the minimal log resolution of singularities. We prove that $\bar E$ has at most six cusps and we establish an effective version of the Zaidenberg Finiteness Conjecture (1994) concerning Eisenbud-Neumann diagrams of $E$. This is done by analysing the Minimal Model Program run for the pair $(X,\frac{1}{2}D)$. Namely, we show that $\mathbb{P}^2\setminus E$ is $\mathbb{C}^{**}$-fibred or for the log resolution of the minimal model the Picard rank, the number of boundary components and their self-intersections are bounded.