Bounded negativity, H-constants and combinatorics
The Bounded Negativity Conjecture (BNC) is the old still outstanding folklore conjecture that for each smooth projective algebraic surface X there is a bound b_X such that C^2 > b_X for each effective reduced divisor C on X. The question recently has become: for which X does BNC hold and what is the bound? BNC is known to sometimes fail in positive characteristic (see Exercise V.1.10 in Hartshorne's Algebraic Geometry), but few failures are known. In particular, no failures are known for rational surfaces nor for any surfaces in characteristic 0. In an effort to better understand how negative C^2 can be, H-constants were introduced at a mini-workshop at Oberwolfach in 2010. In its simplest form, given a reduced singular plane curve C, H(C) = (d^2-\sum_p m_p(C)^2)/r where d=deg(C), the sum is over the singular points of C, m_p(C) is the multiplicity of C at p and r is the number of singular points. If H(C) is bounded below for all C, then BNC holds for rational surfaces, so the question is: how negative can H(C) be? Attempts to address this have established interesting connections to combinatorics and suggest a plausible value for b_X when X is rational.