Q-homology planes satisfying the Negativity Conjecture (Tomasz Pelka, UniBe)

06.12.2019 15:00

A smooth complex algebraic surface S is called a Q-homology plane if H_i(S,Q)=0 for i>0. This holds for example if S is a complement of a rational cuspidal curve in P^2. The geometry of such S is understood unless S is of log general type, in which case the log MMP applied to the log smooth completion (X,D) of S is insufficient. The idea of K. Palka was to study the pair (X,(1/2)D) instead. This approach gives much stronger constraints on the shape of D, and leads to the Negativity Conjecture, which asserts that the Kodaira dimension of K_X+(1/2)D is negative. It is a natural generalization e.g. of the Coolidge-Nagata conjecture about rational cuspidal curves, which was recently proved using these methods by M. Koras and K. Palka.
If this conjecture holds, all Q-homology planes of log general type can be classified. It turns out that, as expected by tom Dieck and Petrie, they are arranged in finitely many discrete families, each obtainable in a uniform way from certain arrangements of lines and conics on P^2. As a consequence, they all satisfy the Strong Rigidity Conjecture of Flenner and Zaidenberg; and their automorphism groups are subgroups of S_3. To illustrate this surprising rigidity, I will show how to construct all rational cuspidal curves (with complements of log general type, satisfying the Negativity Conjecture) inductively, by iterating quadratic Cremona maps. This construction in particular shows that any such curve is uniquely determined, up to a projective equivalence, by the topology of its singular points.

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