Tomasz Pelka
Title: Smooth Q-homology planes satisfying the Negativity Conjecture
Abstract:
Let (X,D) be a smooth completion of a smooth Q-homology plane, that is, an affine surface S whose higher Betti numbers vanish. The Negativity Conjecture of K. Palka asserts that the Kodaira dimension of K_X+(1/2)D is negative, so the minimal model of (X,(1/2)D) is a log Mori fiber space. I will give a global description of smooth Q-homology planes satisfying this conjecture. Assume that S is of log general type, otherwise the geometry is well understood. i will show that, as expected by tom Dieck and Petrie, all such S can be arranged in finitely many discrete families, each obtainable in a uniform way from certain arrangements of planar lines and conics. As a corollary, I will show that these surfaces satisfy the Strong Rigidity Conjecture of Flenner and Zaidenberg; and that their automorphisms groups have order at most six. This desscripton is the main result of my PhD thesis, written under the supervision of K. Palka.