Motivated by the Green-Griffiths-Lang and Lang-Vojta conjectures, it is expected that the algebraic exceptional set of a log-surface $(X,B)$ of log-general type – which parametrizes curves on $(X,B)$ that are not of log-general type – is finite. This conjecture is open even for $\mathbb{P}^2$, which is the case that we will focus on.

It turns out that the fewer irreducible components $B$ has, the more difficult the problem is. To our knowledge, when $B$ has at least three components, the conjecture is true; but very little is known when $B$ has two components. In this talk, we will present a solution to the problem for the case when $B$ has two irreducible components both with degrees at least 5, and its relation to various notions of hyperbolicity in complex and algebraic geometry. This is work in progress.