Tomasz Pelka

Title: Classifying planar rational cuspidal curves

Abstract:
Let E be a rational cuspidal curve on a complex projective plane and 
let (X,D) -> (P^2,E) be the minimal log resolution. Then the surface X\D is 
a Q-homology plane. The Negativity Conjecture asserts that for such 
surfaces \kappa(2K+D)=-\infty, so a minimal model of (X,(1/2)D) is a log Mori fiber 
space. I will explain how to use this condition to classify, up to a projective 
equivalence, all rational cuspidal curves whose complements satisfy the Negativity 
Conjecture. In the most difficult case \kappa(X\D)=2 one obtains several discrete 
families of curves which can be constructed in a uniform way via Cremona maps. 
I will also indicate how this result fits into the upcoming classification of 
Q-homology planes. This is a joint work with Karol Palka.