Piotr Pokora

Title: On the weak bounded negativity conjecture for blow-ups of the complex projective plane

Abstract: 
In the talk, I will present a new approach towards the proof of the weighted 
bounded negativity conjecture. The main result presents an effective lower bound on the 
weighted self-intersection numbers of irreducible and reduced curves, which can be 
formulated as follows.

Let X_{s} be the blowing-up of the complex projective plane along a finite set 
of mutually distinct $s\geq 1$ points. Denote by C a reduced and irreducible curve 
in X_{s}, and by H the pull-back of a general hyperplane section of the complex 
projective plane, then one has $C^{2} \geq - 2s  \cdot (C.H)$.

My talk will be divided into two parts. In the first part, I will present a general 
introduction to the subject with some instructive examples. In the second part, 
I will present a detailed sketch of the proof of the above theorem.

This result is a special case of a much general result which appears in a recent 
joint work with Roberto Laface (Technical University of Munich), Towards the 
weighted bounded negativity conjecture for blow-ups of algebraic surfaces,  
arXiv:1709.04651