Smooth $\mathbb{Q}$-homology planes satisfying the Negativity Conjecture

TL;DR

This paper classifies smooth $Q$-homology planes of log general type satisfying the Negativity Conjecture, showing they form finitely many series derived from line and conic arrangements, and confirms related conjectures on their rigidity and automorphism groups.

Contribution

It provides a complete classification of such surfaces, confirming conjectures about their structure, automorphisms, and rigidity.

Findings

Surfaces form finitely many discrete series from line and conic arrangements.

They satisfy the Rigidity Conjecture of Flenner and Zaidenberg.

Automorphism groups are bounded by six elements.

Abstract

A complex algebraic surface $S$ is a $\mathbb{Q}$-homology plane if $H_{i}(S,\mathbb{Q})=0$ for $i>0$. The Negativity Conjecture of Palka asserts that $\kappa(K_{X}+\tfrac{1}{2}D)=-\infty$, where $(X,D)$ is a log smooth completion of $S$. We give a complete description of smooth $\mathbb{Q}$-homology planes satisfying the Negativity Conjecture. We restrict our attention to those of log general type, as otherwise their geometry is well understood. We show that, as conjectured by tom Dieck and Petrie, they can be arranged in finitely many discrete series, each obtained in a uniform way from an arrangement of lines and conics on $\mathbb{P}^{2}$. We infer that these surfaces satisfy the Rigidity Conjecture of Flenner and Zaidenberg; and a conjecture of Koras, which asserts that $\#\operatorname{Aut}(S)\leq 6$.