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

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 $\kappa (K_X+\frac{1}{2}D)=-\infty$, 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 $\#\mathrm{Aut}(S)\le 6$.

53 pages, 44 figures, to appear in J. London Math. Soc

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• Rational cuspidal curves with four cusps on Hirzebruch surfaces
• Zariski pairs of conic-line arrangements of degrees 7 and 8 via fundamental groups
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